\documentclass[%
version=last,%
a5paper,%
12pt,%
%headings=small,%
%bibliography=totoc,% index=totoc,%
twoside,%
open=any,%
%parskip=half,%  this option takes 2.5% more space than parskip
draft=true,%
%titlepage=false,%
DIV=12,%
headinclude=false,%
pagesize]%
{scrbook}
%{scrartcl}

%\PassOptionsToPackage{pdf}{pstricks} %used for pdflatex
%\newenvironment{postscript}{}{}

%\makeindex
%\usepackage{makeidx}
%\usepackage{showidx}
%  Run texindy -L turkish <filename>.idx on the file

%\usepackage[notcite,notref]{showkeys}

\usepackage{cclicenses}

\usepackage{scrpage2}
\pagestyle{scrheadings}
\clearscrheadings
\ofoot{\pagemark}
\refoot{\leftmark}
\lofoot{\rightmark}
%\cehead{K\"umeler Kuram\i}
%\cohead{\today}

\usepackage{moredefs,lips}

%\newcommand{\letterrefs}[1]{(#1)}
\newcommand{\letterrefs}[1]{}

%\usepackage[turkish]{babel}
\usepackage[polutonikogreek,turkish]{babel}
\newcommand{\eng}[1]{\emph{#1}}
\usepackage{gfsporson}
\newcommand{\gr}[1]{\foreignlanguage{polutonikogreek}{\textporson{#1}}}

%\newcommand{\enquote}[1]{``#1''}
%\usepackage[utf8]{inputenc}

\usepackage{multicol}
\setlength{\multicolsep}{0\baselineskip}
\setlength{\columnseprule}{0.5pt}

\usepackage{verbatim}
 \usepackage{hfoldsty}
\usepackage[perpage,symbol*]{footmisc}

% For symbol table

\usepackage{longtable}

%\makeglossary % This command must be commented when the glossary is to
	      % be printed
\newcommand{\glossaryentry}[2]{#1& #2\\ } % This is used in *.glo
% In the tex file, the \glossary command, with one argument, puts that
% argument as the first argument of \glossaryentry (the second being a
% page number) in the *.glo file.  I am using this *.glo for an index
% of symbols.  Symbols in the Introduction are not included in this
% index, but only because the relevant \glossary{---} commands are commented.

%\usepackage{float}
%\floatstyle{boxed} 
%\restylefloat{figure}

%\renewcommand{\captionformat}{ } % doesn't work with float package

%\newcommand{\nolu}{$\mathscr N^{\text{olu}}$} % or ``numaral\i''
%\newcommand{\nolu}{numaral\i}
\usepackage{ifthen,calc}
\newcounter{rfp}\newcounter{ones}\newcounter{tens}
\usepackage{refcount}

\newcommand{\sayfanumaraya}[1]{%
\setcounterpageref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
   {\value{ones}=6}%
   {ya}%
   {\ifthenelse%
       {\value{ones}=9}%
       {a}%
       {\ifthenelse%
           {\value{ones}=2\or\value{ones}=7}%
           {ye}%
           {\ifthenelse%
               {\value{ones}=0}%
               {\ifthenelse%
                   {\value{tens}=2\or\value{tens}=5}
                   {ye}
                   {e}}
               {e}}}}}
\newcommand{\sayfaya}[1]{sayfa \sayfanumaraya{#1}}

\newcommand{\sayfanumarada}[1]{%
%\setcounter{rfp}{\number\numexpr\getpagerefnumber{#1}}%
\setcounterpageref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
   {\value{ones}=6\or\value{ones}=9}%
   {da}%
   {\ifthenelse%
       {\value{ones}=3\or\value{ones}=4\or\value{ones}=5}%
       {te}%
       {\ifthenelse%
           {\value{ones}=0}%
           {\ifthenelse%
               {\value{tens}=7}%
               {te}
               {\ifthenelse%
                   {\value{tens}=4\or\value{tens}=6}%
                   {ta}
                   {\ifthenelse%
                       {\value{tens}=1\or\value{tens}=3\or\value{tens}=9}%
                       {da}%
                       {de}}}}%
           {de}}}}
\newcommand{\sayfada}[1]{sayfa \sayfanumarada{#1}}
\newcommand{\Sayfada}[1]{Sayfa \sayfanumarada{#1}}

\newcommand{\numaraya}[1]{%
\setcounterref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
   {\value{ones}=6}%
   {ya}%
   {\ifthenelse%
       {\value{ones}=9}%
       {a}%
       {\ifthenelse%
           {\value{ones}=2\or\value{ones}=7}%
           {ye}%
           {\ifthenelse%
               {\value{ones}=0}%
               {\ifthenelse%
                   {\value{tens}=2\or\value{tens}=5}
                   {ye}
                   {e}}
               {e}}}}}


\newcommand{\numarada}[1]{%
%\setcounter{rfp}{\number\numexpr\getpagerefnumber{#1}}%
\setcounterref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
   {\value{ones}=6\or\value{ones}=9}%
   {da}%
   {\ifthenelse%
       {\value{ones}=3\or\value{ones}=4\or\value{ones}=5}%
       {te}%
       {\ifthenelse%
           {\value{ones}=0}%
           {\ifthenelse%
               {\value{tens}=7}%
               {te}
               {\ifthenelse%
                   {\value{tens}=4\or\value{tens}=6}%
                   {ta}
                   {\ifthenelse%
                       {\value{tens}=1\or\value{tens}=3\or\value{tens}=9}%
                       {da}%
                       {de}}}}%
           {de}}}}

\newcommand{\numarayi}[1]{%
\setcounterref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
  {\value{ones}=1\or\value{ones}=5\or\value{ones}=8}%
  {i}%
  {\ifthenelse%
     {\value{ones}=2\or\value{ones}=7}%
     {yi}%
     {\ifthenelse%
        {\value{ones}=3\or\value{ones}=4}%
        {\"u}%
        {\ifthenelse%
           {\value{ones}=6}%
           {y\i}%
           {\ifthenelse%
              {\value{ones}=9}%
              {u}%
              {\ifthenelse%
                 {\value{tens}=7\or\value{tens}=8}%
                 {i}%
                 {\ifthenelse%
                    {\value{tens}=2\or\value{tens}=5}%
                    {yi}%
                    {\ifthenelse%
                       {\value{tens}=1\or\value{tens}=3}%
                       {u}%
                       {\ifthenelse%
                          {\value{tens}=4\or\value{tens}=6\or\value{tens}=9}%
                          {\i}
                          {\"u}}}}}}}}}}

\newcommand{\numaranin}[1]{%
\setcounterref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
  {\value{ones}=1\or\value{ones}=5\or\value{ones}=8}%
  {in}%
  {\ifthenelse%
     {\value{ones}=2\or\value{ones}=7}%
     {nin}%
     {\ifthenelse%
        {\value{ones}=3\or\value{ones}=4}%
        {\"un}%
        {\ifthenelse%
           {\value{ones}=6}%
           {n\i n}%
           {\ifthenelse%
              {\value{ones}=9}%
              {un}%
              {\ifthenelse%
                 {\value{tens}=7\or\value{tens}=8}%
                 {in}%
                 {\ifthenelse%
                    {\value{tens}=2\or\value{tens}=5}%
                    {nin}%
                    {\ifthenelse%
                       {\value{tens}=1\or\value{tens}=3}%
                       {un}%
                       {\ifthenelse%
                          {\value{tens}=4\or\value{tens}=6\or\value{tens}=9}%
                          {\i n}
                          {\"u}}}}}}}}}}

\newcommand{\Teorem}[1]{Teorem \ref{#1}}
\newcommand{\Teoreme}[1]{Teorem \numaraya{#1}}
\newcommand{\Teoremde}[1]{Teorem \numarada{#1}}
\newcommand{\Teoremi}[1]{Teorem \numarayi{#1}}
\newcommand{\Teoremin}[1]{Teorem \numaranin{#1}}

\usepackage{chngcntr}
\counterwithout{figure}{chapter}
\newcommand{\Sekil}[1]{\c Sekil \ref{#1}}
\newcommand{\Sekle}[1]{\c Sekil \numaraya{#1}}
\newcommand{\Sekilde}[1]{\c Sekil \numarada{#1}}

 \renewcommand{\thefootnote}{\fnsymbol{footnote}}

\usepackage[neverdecrease]{paralist}

\usepackage{pstricks,pst-node,pst-tree}

\usepackage{amsmath,amssymb,amsthm,url,upgreek}
\allowdisplaybreaks
%\usepackage{cmbright}
%\renewcommand{\familydefault}{\rmdefault}
%\renewcommand{\rmdefault}{cmr}
%\DeclareMathAlphabet{\mathnormal}{OT1}{cmss}{m}{n}
%\usepackage{unicode-math}
\DeclareMathVersion{false}
\SetSymbolFont{operators}{false}{OT1}{cmbr}{m}{n}
\SetSymbolFont{letters}{false}{OML}{cmbrm}{m}{it}
\SetSymbolFont{symbols}{false}{OMS}{cmbrs}{m}{n}
%\SetSymbolFont{largesymbols}{false}{OMX}{cmex}{m}{n}
\usepackage{relsize}
\newenvironment{falseproof}{%
\begin{quotation}\mathversion{false}\relscale{0.9}}%
{\end{quotation}}

\renewenvironment{quote}{%
\relscale{.90}
\begin{list}{}{%
\setlength{\leftmargin}{0.05\textwidth}
%\setlength{\topsep}{0pt}
\setlength{\topsep}{0.25\baselineskip}
\setlength{\rightmargin}{\leftmargin}
\setlength{\parsep}{\parskip}
}
\item\relax}
{\end{list}}

\renewenvironment{quotation}{%
\begin{list}{}{%
\relscale{.90}
\setlength{\leftmargin}{0.05\textwidth}
%\setlength{\topsep}{0\baselineskip}
\setlength{\topsep}{0.25\baselineskip}
\setlength{\rightmargin}{\leftmargin}
\setlength{\listparindent}{1em}
\setlength{\itemindent}{\listparindent}
\setlength{\parsep}{\parskip}
}
\item\relax}
{\end{list}}

\newtheorem{theorem}{Teorem}
%\newtheorem{proposition}{\"Onerme}
\newtheorem*{corollary}{Sonu\c c}
\newtheorem{lemma}{Lemma}
%\newtheorem*{theorem*}{Teorem}
%\newtheorem{axiom}{Aks\.iyom}
\newtheorem{axiom}{AKS\.IYOM}
%\theoremstyle{definition}
\theoremstyle{remark}

%  The template for the following is taken from
%  http://en.wikibooks.org/wiki/LaTeX/Theorems 
%  The punctuation dot is needed (default is apparently no dot).
%  ``Space after theorem head'' is needed (otherwise errors).
\newtheoremstyle{xca}% name of the style to be used
  {}% measure of space to leave above the theorem. E.g.: 3pt
  {}% measure of space to leave below the theorem. E.g.: 3pt
  {\sffamily}% name of font to use in the body of the theorem
  {}% measure of space to indent
  {\sffamily\itshape}% name of head font
  {.}% punctuation between head and body
  {0.5em}% space after theorem head; " " = normal interword space
  {}% Manually specify head

\theoremstyle{xca}
\newtheorem{xca}{Al\i\c st\i rma}
\renewcommand{\thexca}{\Roman{xca}}

\newcommand{\ktk}[1][]{\begin{xca}Teoremi kan\i tlay\i n.#1\end{xca}}
\newcommand{\ksk}[1][]{\begin{xca}Sonucu kan\i tlay\i n.#1\end{xca}}
%\newcommand{\kok}[1][]{\begin{xca}\"Onermeyi kan\i tlay\i n.#1\end{xca}}
\newcommand{\klk}[1][]{\begin{xca}Lemmay\i\ kan\i tlay\i n.#1\end{xca}}

\usepackage{bm}

\newcommand{\lto}{\Rightarrow}
\newcommand{\liff}{\Leftrightarrow}
\newcommand{\Forall}[1]{\forall{#1}\;}
\newcommand{\Exists}[1]{\exists{#1}\;}
\newcommand{\Zmod}[1]{\Z_{#1}}
\newcommand{\gn}[1]{\ulcorner#1\urcorner}

\newcommand{\norm}[1]{\lVert#1\rVert}
%\newcommand{\denk}{\;\text{denktir}\;}
\newcommand{\denk}{\quad\text{denktir}\quad}
\newcommand{\universe}{\mathbf V}
%\newcommand{\inv}{^{-1}}
\newcommand{\inv}[1]{{#1}^{-1}}
\newcommand{\conv}[1]{\breve{#1}}
\newcommand{\on}{\mathbf{ON}}
\newcommand{\cn}{\mathbf{KN}}
\newcommand{\zf}{\mathrm{ZF}}
\newcommand{\ac}{\mathrm{AC}}
\newcommand{\zfc}{\mathrm{ZFC}}
\newcommand{\ch}{\mathrm{KH}}
\newcommand{\gch}{\mathrm{GKH}}
\newcommand{\ord}{\operatorname{ord}}
\DeclareMathOperator{\cardinal}{kard}
\newcommand{\card}[1]{\cardinal(#1)}
%\newcommand{\cardsum}{+^{\mathrm k}}
\newcommand{\cardsum}{\oplus}
%\newcommand{\cardprod}{\cdot^{\mathrm k}}
\newcommand{\cardprod}{\otimes}
\newcommand{\cof}[1]{\operatorname{kf}(#1)}

\renewcommand{\deg}{\operatorname{der}}
\renewcommand{\max}{\operatorname{maks}}
\renewcommand{\vec}[1]{\bm{#1}}
\newcommand{\letsep}{\quad}

%\newcommand{\symdiff}{\vartriangle}
%\newcommand{\comp}{^{\mathrm c}}

\usepackage{mathrsfs}
\newcommand{\pow}[2][]{\mathscr P^{#1}(#2)}
\newcommand{\powf}[1]{\mathscr P_{\upomega}(#1)}
%\usepackage{MnSymbol}
%\newcommand{\pow}[1]{\powerset(#1)}

\newcommand{\included}{\subseteq}
\newcommand{\nincluded}{\nsubseteq}
\newcommand{\includes}{\supseteq}
\newcommand{\pincluded}{\subset}
\newcommand{\npincluded}{\not\subset}

\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\Qp}{\mathbb Q^+}
\newcommand{\R}{\mathbb R}
\newcommand{\Rp}{\mathbb R^+}
\newcommand{\C}{\mathbb C}


\newcommand{\divides}{\mid}
\newcommand{\ndivides}{\nmid}

\newcommand*{\twoheadrightarrowtail}{\mathrel{\rightarrowtail\kern-1.9ex\twoheadrightarrow}}

\renewcommand{\emptyset}{\varnothing}
\renewcommand{\phi}{\varphi}
\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}
\renewcommand{\setminus}{\smallsetminus}

%\renewcommand{\theequation}{\fnsymbol{equation}}
\renewcommand{\epsilon}{\varepsilon}
\usepackage[all]{xy}

\begin{document}
%\frontmatter
\title{Ordinal Analiz}
\subtitle{Aksiyomatik K\"umeler Kuram\i\ Dersi}
\author{David Pierce}
\date{7 Kas\i m 2019 tasla\u g\i}
\publishers{Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\.Istanbul\\
\url{dpierce@msgsu.edu.tr}\\
\url{mat.msgsu.edu.tr/~dpierce/}}

\maketitle

%\frontmatter

%\selectlanguage{english}

%\mainmatter

%\selectlanguage{turkish}

\tableofcontents

\listoffigures

%\mainmatter

\chapter{K\"umeler kuram\i\ olarak matematik}\label{ch:intro}


Matemati\u gi yaparak zaten k\"umelerin ne oldu\u gunu biliyoruz.
\"Orne\u gin
\begin{compactitem}
  \item
    ger\c cel say\i lar bir $\R$,
    \item
      kesirli say\i lar bir $\Q$,
      \item
        tamsay\i lar bir $\Z$,
        \item
          sayma say\i lar bir $\N$
\end{compactitem}
 k\"umesini
olu\c sturur.

\c Simdi $\R$'den $\R$'ye giden bir $f$ g\"ondermesi verilsin.
\"Orne\u gin
\begin{equation*}
  f(x)=\sin x+x^2-2
\end{equation*}
olsun.  O zaman $f$'nin kendisi,
\begin{equation*}
  \{(x,y)\in\R\times\R\colon y=\sin x+x^2-2\}
\end{equation*}
k\"umesidir.
Burada tan\i ma g\"ore
\begin{gather*}
  (a,b)=\bigl\{\{a\},\{a,b\}\bigr\},\\
  A\times B=\{(x,y)\colon x\in A\And y\in B\}.
\end{gather*}

\begin{theorem}
  T\"um $a$, $b$, $c$, ve $d$ i\c cin
    \begin{equation*}
    (a,b)=(c,d)\iff a=c\And b=d.
  \end{equation*}
\end{theorem}

\begin{proof}
  ($\Rightarrow$).
  $(a,b)=(c,d)$ olsun.  O zaman
  \begin{equation*}
    \bigl\{\{a\},\{a,b\}\bigr\}
    =\bigl\{\{c\},\{c,d\}\bigr\}.
  \end{equation*}
  O halde
  \begin{equation*}
    \{a\}\in\bigl\{\{c\},\{c,d\}\bigr\},
  \end{equation*}
  \.Iki durum vard\i r.
  \begin{enumerate}
  \item
    $\{a\}=\{c\}$ ise $a=c$.
  \item
    $\{a\}=\{c,d\}$ ise $c\in\{a\}$, dolay\i s\i yla tekrar $a=c$.
  \end{enumerate}
  \c Simdi
  \begin{equation*}
    \{a,b\}=\{c,d\}
  \end{equation*}
  olmal\i d\i r,
  dolay\i s\i yla $a=c$ oldu\u gundan $b=d$.

  ($\Leftarrow$).
  A\c c\i kt\i r.
\end{proof}

Her $A$ k\"umesinin altk\"umeleri, $A$'n\i n
\begin{equation*}
  \pow A
\end{equation*}
\textbf{kuvvet k\"umesini} olu\c sturur.
\"Orne\u gin
\begin{align*}
  \{a\}&\included\{a,b\},&
  \{a,b\}&\included\{a,b\}
\end{align*}
oldu\u gundan
\begin{align*}
  \{a\}&\in\pow{\{a,b\}},&
  \{a,b\}&\in\pow{\{a,b\}},
\end{align*}
dolay\i s\i yla
\begin{equation*}
  \bigl\{\{a\},\{a,b\}\bigr\}\included\pow{\{a,b\}},
\end{equation*}
yani
\begin{equation*}
  (a,b)\in\pow{\pow{\{a,b\}}}.
\end{equation*}

\begin{theorem}\label{thm:x}
  $A$ ve $B$ k\"ume ise
  \begin{equation*}
    A\times B\included\pow{\pow{A\cup B}}.
  \end{equation*}
\end{theorem}

\ktk

\c Simdi $\R$'den $\pow{\R}$'ye giden,
\begin{equation}\label{eqn:g}
  g(a)=\{x\in\Q\colon x<a\}
\end{equation}
tan\i m\i n\i\ sa\u glayan bir $g$ g\"ondermesi vard\i r.
$\Q$ $\R$'de \emph{yo\u gun} oldu\u gundan
$g$ birebirdir.
Ayr\i ca her $g(a)$ k\"umesi,
$\R$'nin
\begin{gather}\label{eqn:0AR}
  \emptyset\pincluded A\pincluded\R,\\\label{eqn:A}
    b\in A\And c<b\implies c\in A
\end{gather}
ko\c sullar\i n\i\ sa\u glayan bir $A$ altk\"umesidir.

Tam tersine $A$,
elemanlar\i\ ger\c cel say\i\ olan,
verilen \eqref{eqn:0AR} ve \eqref{eqn:A}
ko\c sullar\i n\i\ sa\u glayan bir k\"ume olsun.
O zaman
\begin{compactitem}
\item
  \eqref{eqn:0AR} sayesinde
  $\R\setminus A$ fark\i n\i n bir $c$ eleman\i\ vard\i r, ve
\item
  \eqref{eqn:A} ko\c sulunun
  \begin{equation*}
    c\in\R\setminus A\And b\in A\implies b<c
  \end{equation*}
  kar\c s\i t tersine g\"ore
$c$ $A$'n\i n bir \"usts\i n\i r\i d\i r.
\end{compactitem}
O zaman
\begin{compactitem}
  \item
$A$ bo\c s olmad\i g\i ndan,
\item
$\R$ do\u grusal s\i ralanm\i\c s bir k\"ume olarak
\emph{tam} oldu\u gundan
\end{compactitem}
$A$'n\i n en k\"u\c c\"uk \"usts\i n\i r\i\ vard\i r.
\c Simdi ayr\i ca
\begin{equation}\label{eqn:open}
  \sup A\notin A
\end{equation}
olsun.
O zaman
\begin{equation*}
  g(\sup A)=A.
\end{equation*}
Genel bir teorem vard\i r.

\begin{theorem}
  $(A,<)$,
  u\c csuz yo\u gun do\u grusal s\i ralanm\i\c s bir k\"ume olsun,
  ve $A$'n\i n
  \begin{gather*}
    X\neq\emptyset,\qquad
    X\neq A,\\
    \Forall x\Forall y(y\in X\land x<y\lto x\in X),\\
    \Forall x\Exists y(x\in X\lto x<y\land y\in X)
  \end{gather*}
  c\"umlelerini sa\u glayan $X$ altk\"umeleri $A^*$ k\"umesini olu\c stursun.
  O zaman
  \begin{compactenum}[(a)]
    \item
  $(A^*,\pincluded)$ yap\i s\i\ da
      u\c csuz yo\u gun do\u grusal s\i ralanm\i\c s bir k\"umedir,
    \item
      oraya $(A,<)$
      \begin{equation*}
        x\mapsto\{y\in A\colon y<x\}
      \end{equation*}
      g\"ondermesi alt\i nda g\"om\"ul\"ur,
      \item
        $(A^*,\pincluded)$ tamd\i r,
      \item
        E\u ger $B$, $A^*$'\i n bo\c s olmayan,
        \"usts\i n\i r\i\ olan bir altk\"umesi ise,
        o zaman
        \begin{equation*}
          \sup B=\bigcup B.
        \end{equation*}
  \end{compactenum}
\end{theorem}

\ktk

\c Simdi
$\R$'nin
\eqref{eqn:0AR},
\eqref{eqn:A}, ve \eqref{eqn:open} ko\c sullar\i n\i\ sa\u glayan
$A$ k\"umeleri
 $\mathscr S$ k\"umesini olu\c stursun.
O zaman tan\i m\i\ \eqref{eqn:g} olan $g$ g\"ondermesi,
$\R$'den $\mathscr S$'ye giden bir e\c slemedir.
Ayr\i ca $\R$'yi kullan\i lmadan $\mathscr S$ tan\i mland\i\u g\i ndan
$\R$, $\mathscr S$ olarak tan\i mlanabilir.
K\i saca her ger\c cel say\i,
elemanlar\i\ kesirli say\i\ olan bir k\"ume olarak tan\i mlanabilir.
\"Ozel olarak
\begin{equation*}
  \R\included\pow{\Q}.
\end{equation*}

Ayr\i ca tan\i ma g\"ore
\begin{equation*}
  \Q=\left\{\frac xy\colon(x,y)\in\Z\times(\Z\setminus\{0\})\right\},
\end{equation*}
ve burada
\begin{equation*}
  \frac ab=\frac cd\iff ad=bc.
\end{equation*}
Bir $a/b$ kesirli say\i s\i n\i n iyitan\i mlanmas\i\ i\c cin
$\Z\times(\Z\setminus\{0\})$ \c carp\i m\i nda
\begin{equation*}
  (x,y)\mathrel B(z,w)\iff xw=yz
\end{equation*}
kural\i\
taraf\i ndan tan\i mlanm\i\c s $B$ ba\u g\i nt\i s\i,
bir denklik ba\u g\i nt\i s\i\ olmal\i d\i r.
Bu durumda
\begin{equation*}
  \frac ab=\{(x,y)\in\Z\times(\Z\setminus\{0\})\colon(a,b)\mathrel B(x,y)\},
\end{equation*}
dolay\i s\i yla \Teorem{thm:x} sayesinde
\begin{gather*}
  \frac ab\included\pow{\pow{\Z}},\\
  \Q\included\pow{\pow{\pow{\Z}}}.
\end{gather*}
Benzer \c sekilde
\begin{equation*}
  a-b=\{(x,y)\in\N\times\N\colon a+y=b+x\}
\end{equation*}
olmak \"uzere
\begin{equation*}
  \Z=\{x-y\colon(x,y)\in\N\times\N\},
\end{equation*}
dolay\i s\i yla
\begin{gather*}
  \Z\included\pow{\pow{\pow{\N}}},\\
  \Q\included\pow{\pow{\pow{\pow{\pow{\pow{\N}}}}}},\\
  \R\included\pow{\pow{\pow{\pow{\pow{\pow{\pow{\N}}}}}}}.
\end{gather*}
K\"ume olarak $\N$'nin elemanlar\i\ nas\i l anla\c s\i labilir?


\chapter{Do\u gal say\i lar}\label{ch:ax} % check label


\section{S\i n\i flar ve k\"umeler}\label{sect:sets-classes}

Her \emph{k\"ume} bir \emph{s\i n\i ft\i r,}
ama her s\i n\i f bir k\"ume de\u gildir.

Tan\i ma g\"ore bir \textbf{s\i n\i f,}
\emph{tek serbest de\u gi\c skenli bir form\"ul} taraf\i ndan
tan\i mlan\i r.
Bir s\i n\i f\i n \textbf{elemanlar\i} vard\i r,
ve bunlar,
tan\i mlayan form\"ul\"un sa\u glayanlar\i d\i r.

Baz\i\ s\i n\i flar, k\"umedir.
K\"umelerin \"ozellikleri
bir tan\i m de\u gil,
aksiyomlar taraf\i ndan verilecektir.
\c Simdilik, e\u ger $a$ bir k\"ume ise,
o zaman
\begin{equation*}
  x\in a
\end{equation*}
ifadesi,
serbest de\u gi\c skeni $x$ olan bir form\"uld\"ur,
ve bu form\"ul, s\i n\i f olarak $a$'y\i\ tan\i mlar.

E\u ger $\phi$
\label{phi}%
serbest de\u gi\c skeni $x$ olan bir form\"ul ise,
o zaman $\phi$'nin tan\i mlad\i\u g\i\ s\i n\i f\i,
\begin{equation*}
\{x\colon\phi\}
\end{equation*}
olarak yaz\i l\i r.
E\u ger bu s\i n\i f i\c cin $\bm A$ harf\/ini yazarsak,
o zaman her $b$ k\"umesi i\c cin
$\phi(b)$ c\"umlesinin yerine $b\in\bm A$ ifadesini yazabiliriz:
k\i saca
\begin{equation*}
\phi(b)\iff  b\in\bm A.
\end{equation*}
Her taraf\i na g\"ore $b$, $\bm A$'n\i n eleman\i d\i r.

Form\"uller ve onlar\i n serbest de\u gi\c skenleri,
\ref{mantik} Eki'nde tan\i mlan\i r.
S\i n\i flar b\"uy\"uk siyah harfler ile g\"osterece\u giz;
k\"u\c c\"uk harfler
her zaman k\"ume olacakt\i r.
B\"uy\"uk normal harfler de k\"ume olabilir.

Bizim i\c cin her s\i n\i f\i n her eleman\i\ bir k\"ume olacakt\i r.
K\"ume olmayan bir s\i n\i f, bir s\i n\i f\i n eleman\i\ olamaz.
Bundan dolay\i, $x\in a$ veya $x\in\bm B$ gibi bir form\"ulde,
$x$ de\u gi\c skeninin yerine $\bm C$ gibi b\"uy\"uk siyah bir harf konulamaz.

Elemanlar\i\ ayn\i\ olan s\i n\i flar da ayn\i d\i r,
ama bu ayn\i l\i k, \textbf{e\c sitlik} olarak yaz\i l\i r.
(\"Oklid'de e\c sitlik, ayn\i l\i k de\u gildir.
\"Orne\u gin
ikizkenar bir \"u\c cgenin iki e\c sit kenar\i\ vard\i r,
ama bu kenarlar birbiriyle ayn\i\ de\u gildir.)
B\"oylece
\begin{equation*}
\bm A=\bm B\iff\Forall x(x\in\bm A\liff x\in\bm B).  
\end{equation*}
\"Ozel olarak
\begin{equation*}
  a=\{x\colon x\in a\}.
\end{equation*}

\begin{theorem}[Russell Paradoksu]\label{thm:Russell}
$\{x\colon x\notin x\}$
s\i n\i f\i\
bir k\"ume de\u gildir.
\end{theorem}

\begin{proof}
  $x\notin x$ form\"ul\"u $\phi$ olarak yaz\i ls\i n.
  M\"umk\"unse $\{x\colon\phi\}=a$ olsun.
O zaman her $b$ k\"umesi i\c cin
\begin{equation*}
b\in a\liff\phi(b).
\end{equation*}
\"Ozel olarak
$a\in a\liff\phi(a)$,
yani
\begin{equation*}
a\in a\liff a\notin a;
\end{equation*}
ama bu bir \c celi\c skidir.
Bu \c sekilde $\{x\colon x\notin x\}$ s\i n\i f\i,
bir $a$ k\"umesine e\c sit olamaz.
\end{proof}

Tan\i ma g\"ore
\begin{gather*}
  \universe=\{x\colon x\in x\lor x\notin x\},\\
  \emptyset=\{x\colon x\in x\land x\notin x\},\\
  \bm A\cup\bm B=\{x\colon x\in\bm A\lor x\in\bm B\},\\
  \bm A\cap\bm B=\{x\colon x\in\bm A\land x\in\bm B\},\\
  \bm A\setminus\bm B=\{x\colon x\in\bm A\land x\notin\bm B\}.
\end{gather*}
O zaman
\begin{align*}
  \bm B\cup\bm A&=\bm A\cup\bm B,&
  \bm A\cup\emptyset&=\bm A,&
  \bm A\cup\universe&=\universe,\\
  \bm B\cap\bm A&=\bm A\cap\bm B,&
  \bm A\cap\emptyset&=\emptyset,&
  \bm A\cap\universe&=\bm A,
\end{align*}
ve ayr\i ca
\begin{align*}
  \bm A\setminus\emptyset&=\bm A,&
  \bm A\setminus\universe&=\emptyset.
\end{align*}
Her $a$ k\"umesi i\c cin
tan\i ma g\"ore
\begin{equation*}
  \{a\}=\{x\colon x=a\};
\end{equation*}
burada $x=a$ ifadesi,
$\Forall y(y\in x\liff y\in a)$
form\"ul\"u i\c cin bir k\i saltmad\i r.
Benzer \c sekilde
\begin{gather*}
  \{a,b\}=\{x\colon x=a\lor x=b\},\\
  \{a,b,c\}=\{x\colon x=a\lor x=b\lor x=c\},\\
  \makebox[7cm]{\dotfill}
\end{gather*}


\begin{axiom}[Bo\c s K\"ume]
  $\emptyset$ s\i n\i f\i\
bir k\"umedir.
\end{axiom}

Bo\c s k\"ume $0$ olarak da yaz\i l\i r.

\begin{axiom}[Biti\c stirme]
  T\"um $a$ ve $b$ k\"umeleri i\c cin
  \begin{equation*}
    a\cup\{b\}
  \end{equation*}
  s\i n\i f\i\
  bir k\"umedir.
\end{axiom}

Sonu\c c olarak her $\{a\}$, $\{a,b\}$,
$\{a,b,c\}$, veya \lips s\i n\i f\i\ bir k\"umedir.
\"Ozellikle
\begin{align*}
  &\{0\},&
  &\bigl\{0,\{0\}\bigr\},&
  &\Bigl\{0,\{0\},\bigl\{0,\{0\}\bigr\}\Bigr\},&
  &\dots
\end{align*}
s\i n\i flar\i n\i n her biri, bir k\"umedir.
Bu k\"umeler, s\i ras\i yla $1$, $2$, $3$, \lips,
olarak yaz\i l\i r.

Tan\i ma g\"ore her $a$ k\"umesi i\c cin
\begin{equation*}
  a'=a\cup\{a\}
\end{equation*}
olsun.
O zaman Biti\c stirme Aksiyomu'na g\"ore $a'$ s\i n\i f\i,
bir k\"umedir.
Bu k\"umeye $a$'n\i n \textbf{ard\i l\i} densin.
\c Simdi
\begin{align*}
  1&=0',&
  2&=1',&
  3&=2',&
  &\dots
\end{align*}
$0$'\i\ i\c ceren,
her eleman\i n\i n ard\i l\i n\i\ da i\c ceren bir s\i n\i fa
\textbf{t\"umevar\i ml\i} densin.
\"Oyleyse bir $\bm A$ s\i n\i f\i\ t\"umevar\i ml\i d\i r
ancak ve ancak
  \begin{compactenum}[1)]
  \item
    $0\in\bm A$,
  \item
    $\Forall x(x\in\bm A\lto x'\in\bm A)$.
  \end{compactenum}
O zaman $\universe$ t\"umevar\i ml\i d\i r.
  
\begin{axiom}[Sonsuzluk]
T\"umevar\i ml\i\ bir k\"ume vard\i r.
\end{axiom}



\section{Alts\i n\i flar}

E\u ger bir $\bm A$ s\i n\i f\i n\i n her eleman\i\
bir $\bm B$ s\i n\i f\i n\i n bir eleman\i\ ise,
o zaman $\bm A$ $\bm B$'nin bir \textbf{alts\i n\i f\i d\i r}
ve
\begin{equation*}
  \bm A\included\bm B
\end{equation*}
ifadesini yazar\i z.
Bu ifade,
\begin{equation*}
  \Forall x(x\in\bm A\lto x\in\bm B)
\end{equation*}
c\"umlesinin bir k\i saltmas\i d\i r.
O zaman
\begin{equation}\label{eqn:A=B}
  \bm A=\bm B\liff\bm A\included\bm B\land\bm B\included\bm A.
\end{equation}
Ayr\i ca
\begin{equation*}
  \bm A\included\bm B\land
  \bm B\included\bm C\lto
  \bm A\included\bm C,
\end{equation*}
dolay\i s\i yla tan\i ma g\"ore
\begin{equation*}
  \bm A\included\bm B\included\bm C\iff
  \bm A\included\bm B\land
  \bm B\included\bm C.
\end{equation*}
S\"ozc\"uklerde bir s\i n\i f
\begin{compactitem}
\item
  elemanlar\i n\i\ \textbf{i\c cerir,}
\item
  alts\i n\i flar\i n\i\ \textbf{kapsar.}
\end{compactitem}
O zaman $a'$ $a$'y\i\ hem i\c cerir hem de kapsar:
\begin{align*}
  a&\in a',&
  a&\included a'.
\end{align*}
E\u ger bir s\i n\i f\i n bir alts\i n\i f\i\ bir k\"ume ise,
bu k\"ume s\i n\i f\i n bir \textbf{altk\"umesidir.}



\begin{axiom}[Ay\i rma]
Her k\"umenin her alts\i n\i f\i\ bir k\"umedir.
\end{axiom}

Bu \c sekilde her $a$ k\"umesi ve $\{x\colon\phi(x)\}$ s\i n\i f\i\ i\c cin
\begin{equation*}
\{x\colon x\in a\land\phi(x)\}
\end{equation*}
s\i n\i f\i\ bir k\"umedir.
Bu k\"ume
\begin{equation*}
\{x\in a\colon\phi(x)\}
\end{equation*}
olarak yaz\i labilir.

Her s\i n\i f $\universe$'nin bir alts\i n\i f\i\ oldu\u gundan
\Teorem{thm:Russell} sayesinde $\universe$ bir k\"ume de\u gildir.

Tan\i ma g\"ore her $\bm A$ s\i n\i f\i\ i\c cin
\begin{gather*}
  \bigcap\bm A=\{x\colon\Forall y(y\in\bm A\lto x\in y)\},\\
  \bigcup\bm A=\{x\colon\Exists y(y\in\bm A\land x\in y)\}.
\end{gather*}
Burada
\begin{compactitem}
  \item
    $\bigcap\bm A$ $\bm A$'n\i n \textbf{kesi\c simidir,}
    \item
  $\bigcup\bm A$ $\bm A$'n\i n \textbf{bile\c simidir.}
\end{compactitem}
\"Orne\u gin
\begin{align*}
  \bigcap\{a,b\}&=a\cap b,&\bigcup\{a,b\}&=a\cup b,\\
  \bigcap\{a\}&=a,&\bigcup\{a\}&=a,\\
  \bigcap\emptyset&=\universe,&\bigcup\emptyset&=\emptyset.
\end{align*}
Ayr\i ca
\begin{gather}\notag
  \bm A\included\bm B\lto\bigcap\bm B\included\bigcap\bm A
  \land\bigcup\bm A\included\bigcup\bm B,\\\label{eqn:a-in-B}
  a\in\bm B\lto\bigcap\bm B\included a\included\bigcup\bm B.
\end{gather}
Ay\i rma Aksiyomu'ndan $\bm A$ bo\c s de\u gilse $\bigcap\bm A$ bir k\"umedir.

Tan\i ma g\"ore t\"umevar\i ml\i\ k\"umelerin olu\c sturdu\u gu
s\i n\i f\i n kesi\c simi,
\textbf{do\u gal say\i lar\i} i\c cerir,
ve bu kesi\c simi i\c cin $\upomega$ (omega) harf\/ini yazar\i z.
B\"oylece
\begin{equation*}
  \upomega=\bigcap\{x\colon 0\in x\land\Forall y(y\in x\lto y'\in x)\}.
\end{equation*}
Sonsuzluk ve Ay\i rma aksiyomlar\i\ sayesinde $\upomega$ bir k\"umedir.

\begin{theorem}[T\"umevar\i m]\label{thm:ind}
  $A$, $\upomega$'n\i n \"oyle altk\"umesi olsun ki
  \begin{enumerate}[1)]
    \item
    $0\in A$ olsun ve
  \item
    her $n$ do\u gal say\i s\i\ i\c cin
      $n\in A\lto n'\in A$ olsun.
  \end{enumerate}
  O zaman $A=\upomega$.
  K\i saca
  $\upomega$'n\i n tek t\"umevar\i ml\i\ altk\"umesi, kendisidir.
\end{theorem}

\begin{proof}
  Varsay\i ma g\"ore $A$,
  $\upomega$'n\i n t\"umevar\i ml\i\ bir altk\"umesidir.
  \"Oyleyse
  \eqref{eqn:a-in-B} sayesinde
  \begin{align*}
    A&\included\upomega,&
    \upomega&\included A,
  \end{align*}
  dolay\i s\i yla \eqref{eqn:A=B} sayesinde $A=\upomega$.
\end{proof}

\begin{theorem}\label{thm:0-n'}
  Her do\u gal say\i n\i n ya $0$
  ya da do\u gal bir say\i n\i n ard\i l\i\ oldu\u gunu g\"osterin.
\end{theorem}

\ktk

\begin{theorem}\label{thm:o-trans}
  Her do\u gal say\i n\i n her eleman\i,
  do\u gal say\i d\i r.
\end{theorem}

\ktk

Her k\"ume, kendinin altk\"umesidir; k\"umenin di\u ger altk\"umeleri,
k\"umenin \textbf{\"ozaltk\"umeleridir.}
K\i saca
\begin{equation*}
  a\pincluded b\iff a\included b\land a\neq b.
\end{equation*}
O zaman
$\pincluded$ ba\u g\i nt\i s\i\
\textbf{yans\i mas\i z} ve
\textbf{ge\c ci\c slidir,} yani
\begin{gather*}
  a\npincluded a,\\
  a\pincluded b\land b\pincluded c\lto a\pincluded c.
\end{gather*}
K\i saca $\pincluded$, bir \textbf{s\i ralamad\i r}.


\begin{theorem}\label{thm:n-trans}
  Her do\u gal say\i n\i n her eleman\i,
  say\i n\i n bir \"ozaltk\"umesidir:
  E\u ger $n\in\upomega$ ise, o zaman
  \begin{equation*}
    k\in n\lto k\pincluded n.
  \end{equation*}
\end{theorem}

\begin{proof}
  T\"umevar\i m kullanaca\u g\i z.
  \begin{asparaenum}
    \item
      $0$'n\i n eleman\i\ olmad\i\u g\i ndan
      $0$'\i n her eleman\i\ bir \"ozaltk\"umedir.
    \item
      Bir $m$ do\u gal say\i s\i\ i\c cin
      t\"umevar\i m hipotezi olarak
      $m$'nin her eleman\i\
      bir \"ozaltk\"ume olsun.
      Bu durumda $m\npincluded m$ oldu\u gundan
        $m\notin m$,
      dolay\i s\i yla
      \begin{equation}\label{eqn:m-m'}
        m\pincluded m'.
      \end{equation}
      \c Simdi $k\in m'$ olsun.
      O zaman
      \begin{equation*}
        k\in m\lor k=m.
      \end{equation*}
      \begin{compactenum}
      \item
        E\u ger $k\in m$ ise hipotezden $k\pincluded m$.
      \item
        E\u ger $k=m$ ise $k\included m$.
      \end{compactenum}
      Her durumda \eqref{eqn:m-m'} sayesinde
      \begin{equation*}
        k\pincluded m'.
      \end{equation*}
  B\"oylece t\"umevar\i m tamamd\i r.\qedhere
  \end{asparaenum}
\end{proof}

\begin{corollary}
  Ard\i llar\i\ ayn\i\ olan do\u gal say\i lar da ayn\i d\i r:
  $\upomega$'da
  \begin{equation*}
    k'=n'\lto k=n.
  \end{equation*}
\end{corollary}

\ksk

\begin{compactenum}[1)]
\item
  \Teorem{thm:ind},
\item
  \Teoremin{thm:n-trans} sonucu,
  ve
\item
  $0$'\i n ard\i l olmad\i\u g\i na
\end{compactenum}
\textbf{Peano Aksiyomlar\i} denir.
Bunlardan do\u gal say\i lar\i n
matematikte kullan\i lan t\"um \"ozellikleri kan\i tlanabilir.
Do\u gal say\i lar\i n k\"umeler kuram\i ndan gelen tan\i m\i n\i\ kullanarak
\"ozellikleri kan\i tlamak daha kolayd\i r.

\begin{theorem}\label{thm:inc-in}
  E\u ger $n\in\upomega$ ise,
  o zaman her $k$ do\u gal say\i s\i\ i\c cin
  \begin{equation*}
    k\pincluded n\lto k\in n.
  \end{equation*}
\end{theorem}

\begin{proof}
  T\"umevar\i m kullanaca\u g\i z.
  \begin{asparaenum}
    \item
      Her $k$ do\u gal say\i s\i\ i\c cin $k\npincluded 0$,
      dolay\i s\i yla
      \begin{equation*}
        k\pincluded0\lto k\in0.
      \end{equation*}
    \item
      Bir $m$ do\u gal say\i s\i\ i\c cin
      t\"umevar\i m hipotezi olarak
   her $k$ do\u gal say\i s\i\ i\c cin
  \begin{equation*}
    k\pincluded m\liff k\in m
  \end{equation*}
  olsun.
    E\u ger $k\pincluded m'$ ise, o zaman $k\included m$ veya $m\in k$.
    Son durum imk\^ans\i zd\i r \c c\"unk\"u $m\in k$ ise,
    \Teoremde{thm:n-trans}n
    \begin{equation*}
      m\pincluded k\pincluded m',
    \end{equation*}
    ki bu imk\^ans\i zd\i r.
    Sonu\c c olarak $k\included m$.
    \begin{compactitem}
      \item
        E\u ger $k=m$ ise, o zaman $k\in m'$.
      \item
        E\u ger $k\pincluded m$ ise, o zaman $k\in m$,
        dolay\i s\i yla $k\in m'$.
    \end{compactitem}
    T\"umevar\i m tamamd\i r.\qedhere
  \end{asparaenum}
\end{proof}

Sonu\c c olarak $\upomega$'da $\in$ ve $\pincluded$,
ayn\i\ ba\u g\i nt\i d\i r.
\"Ozel olarak $\upomega$'da $\in$ ba\u g\i nt\i s\i\
bir s\i ralamad\i r.

  
\begin{theorem}
  T\"um $k$ ve $n$ do\u gal say\i lar\i\ i\c cin
  \begin{equation*}
    k\included n\lor n\pincluded k.
  \end{equation*}
\end{theorem}

\begin{proof}
  \begin{asparaenum}
    \item
    Her $k$ i\c cin $0\included k$, dolay\i s\i yla
    \begin{equation*}
      k\included 0\lor 0\pincluded k.
    \end{equation*}
    \item
    $\upomega$'da bir $m$ i\c cin,
    her $k$ i\c cin,
    \begin{equation*}
      k\included m\lor m\pincluded k
    \end{equation*}
    olsun, ama bir $k$ i\c cin $k\nincluded m'$ olsun.
    O zaman
    \begin{align*}
      k&\neq m',&
      k&\nincluded m,
    \end{align*}
    dolay\i s\i yla, hipotez sayesinde, $m\pincluded k$.
    Ayr\i ca \Teoremde{thm:inc-in}n $m\in k$, dolay\i s\i yla
    \begin{equation*}
      m'\pincluded k.\qedhere
    \end{equation*}
  \end{asparaenum}
\end{proof}

Sonu\c c olarak $\upomega$'da $\pincluded$ veya $\in$ s\i ralamas\i,
\textbf{do\u grusal} bir s\i ralamad\i r.

\begin{theorem}\label{thm:n-wo}
  $\in$ s\i ralamas\i na g\"ore
  her do\u gal say\i n\i n bo\c s olmayan her altk\"umesinin
  en k\"u\c c\"uk eleman\i\ vard\i r,
  ve bu eleman, altk\"umenin kesi\c simidir.
\end{theorem}

\begin{proof}
  \begin{asparaenum}
  \item
    $0$'\i n bo\c s olmayan hi\c c altk\"umesi olmad\i\u g\i ndan
    iddia $0$ i\c cin a\c sik\^ar bir \c sekilde do\u grudur.
  \item
    Bir $m$ i\c cin iddia do\u gru olsun, ve
    \begin{equation*}
      0\pincluded A\included m'
    \end{equation*}
    olsun.
    \.Iki durum vard\i r.
    \begin{compactenum}
      \item
      E\u ger $A=\{m\}$ ise,
      o zaman $m$,
      a\c sik\^ar bir \c sekilde
      $A$'n\i n di\u ger elemanlar\i n\i n eleman\i d\i r,
      \c c\"unk\"u ba\c ska eleman yoktur.
      \item
        Di\u ger durumda hipoteze g\"ore
        $\bigcap(A\setminus\{m\})$ kesi\c simi,
      $A\setminus\{m\}$ fark\i n\i n en k\"u\c c\"uk eleman\i d\i r.
      Ayr\i ca
      \begin{equation*}
        A\setminus\{m\}\included m,
      \end{equation*}
      dolay\i s\i yla
        $\bigcap(A\setminus\{m\})$
      $A$'n\i n en k\"u\c c\"uk eleman\i d\i r.
      \qedhere
    \end{compactenum}
  \end{asparaenum}
\end{proof}

\begin{corollary}%\label{cor:o-wo}
  $\in$ s\i ralamas\i na g\"ore
  $\upomega$'n\i n bo\c s olmayan her altk\"umesinin
  en k\"u\c c\"uk eleman\i\ vard\i r,
  ve bu eleman, altk\"umenin kesi\c simidir.  
\end{corollary}

\ksk

K\i saca $\upomega$'da $\in$ veya $\pincluded$
do\u grusal s\i ralamas\i,
\textbf{iyis\i ralamad\i r.}
Ayr\i ca her do\u gal say\i da da $\in$ bir iyis\i ralamad\i r.

\begin{xca}[G\"u\c cl\"u T\"umevar\i m]
  $A$, $\upomega$'n\i n \"oyle altk\"umesi olsun ki
  \begin{itemize}
  \item
    her $n$ do\u gal say\i s\i\ i\c cin
    \begin{equation*}
      n\included A\lto n\in A
    \end{equation*}
    olsun.
    (K\i saca $A$, kapsad\i\u g\i\ her do\u gal say\i y\i\ i\c cersin.)
  \end{itemize}
  $A$'n\i n $\upomega$ oldu\u gunu kan\i tlay\i n.
\emph{\.Ipucu}:
  \"Once $\bm A$'n\i n her do\u gal say\i y\i\
  kapsad\i\u g\i n\i\ kan\i tlay\i n.
\end{xca}

\section{Ba\u g\i nt\i lar ve G\"ondermeler}

Bildi\u gimiz gibi
\begin{compactitem}
\item
  $=$ yans\i mal\i, simetrik, ve ge\c ci\c sli bir ba\u g\i nt\i d\i r;
\item
  $\included$ yans\i mal\i\ ve ge\c ci\c sli bir ba\u g\i nt\i d\i r;
\item
  $\pincluded$ yans\i mas\i z ve ge\c ci\c sli bir ba\u g\i nt\i d\i r,
  dolay\i s\i yla bir s\i ralamad\i r;
\item
  $\upomega$'da $\pincluded$ do\u grusal bir s\i ralamad\i r
  ve $\in$ ile ayn\i d\i r.
\end{compactitem}
Genelde bir \textbf{ba\u g\i nt\i,}
iki serbest de\u gi\c skeni olan bir form\"ul taraf\i ndan tan\i mlan\i r.
\"Orne\u gin $=$, $\included$, $\pincluded$, ve $\in$,
s\i ras\i yla
\begin{gather*}
  \Forall z(z\in x\liff z\in y),\\
  \Forall z(z\in x\lto z\in y),\\
  \Forall z(z\in x\lto z\in y)\land\Exists z(z\notin x\land z\in y),\\
  x\in y
\end{gather*}
form\"ulleri taraf\i ndan tan\i mlan\i r.
E\u ger bir $\phi$ form\"ul\"u bir $\bm R$ ba\u g\i nt\i s\i n\i\ tan\i mlarsa,
o zaman $\phi(a,b)$ c\"umlesini
\begin{equation*}
  a\mathrel{\bm R}b
\end{equation*}
olarak yazabiliriz.


\c Simdi $\bm R$ bir ba\u g\i nt\i\ ve $\bm D$ bir s\i n\i f olsun.
E\u ger $\bm D$'nin her $a$ eleman\i\ i\c cin
ve b\"ut\"un $b$ ve $c$ k\"umeleri i\c cin
\begin{equation*}
  a\mathrel{\bm R}b
  \land
  a\mathrel{\bm R}c
  \lto
  b=c
\end{equation*}
ise, o zaman $\bm D$'de $\bm R$ bir \textbf{g\"ondermeyi} tan\i mlar.
Bu g\"ondermeye $\bm F$ densin;
o zaman $\bm F$'nin \textbf{tan\i m s\i n\i f\i} $\bm D$'dir.
E\u ger
\begin{equation*}
  a\in\bm D\land a\mathrel{\bm R}b
\end{equation*}
ise, o zaman
\begin{equation*}
  \bm F(a)=b
\end{equation*}
yaz\i labilir;
ayr\i ca $\bm F$'nin yerine
\begin{equation*}
  x\mapsto\bm F(x)
\end{equation*}
yaz\i labilir.
\"Orne\u gin $\universe$'de
  $x\mapsto x'$
g\"ondermesi vard\i r,
ve bu g\"ondermeyi
\begin{equation*}
  \Forall z\bigl(z\in y\liff z\in x\lor\Forall w(w\in z\liff w\in x)\bigr)
\end{equation*}
form\"ul\"u,
k\i saca
$\Forall z(z\in y\liff z\in x\lor z=x)$,
tan\i mlar.


\begin{axiom}[Yerle\c stirme]
  Her $\bm F$ g\"ondermesi i\c cin,
  $\bm F$'nin tan\i m s\i n\i f\i n\i n her $a$ altk\"umesi i\c cin
\begin{equation*}
\{y\colon\Exists x(\bm F(x)=y\land x\in a)\}
\end{equation*}
s\i n\i f\i\ bir k\"umedir.
\end{axiom}

Aksiyomda verilen k\"ume
\begin{align*}
  &\{\bm F(x)\colon x\in a\},&
  &\bm F[a]
\end{align*}
ifadelerinin biri ile g\"osterilebilir.


  \section{Do\u gal Say\i larda \"Ozyineleme}


  \begin{theorem}[\"Ozyineleme]\label{thm:rec}
    Bir $A$ k\"umesi i\c cin
    \begin{compactenum}[1)]
    \item
      $b\in A$,
    \item
      $f\colon A\to A$ 
    \end{compactenum}
    olsun.
    O zaman $\upomega$'dan $A$'ya giden bir ve tek bir $g$ g\"ondermesi i\c cin
    \begin{compactenum}[1)]
    \item
      $g(0)=b$,
    \item
      her $k$ do\u gal say\i s\i\ i\c cin $g(k+1)=f(g(k))$.
    \end{compactenum}
    \Sekle{fig:rec} bak\i n.
    \begin{figure}
      \relscale{1.1}
      \begin{equation*}
        \xymatrix@=2cm{\{0\}\ar[d]_g\ar[r]^{\included}&\upomega\ar[d]_g\ar[r]^{x\mapsto x+1}&\upomega\ar[d]^g\\
          \{b\}\ar[r]_{\included}&A\ar[r]_f&A}
      \end{equation*}
      \caption{\"Ozyineleme}\label{fig:rec}
    \end{figure}
  \end{theorem}

  \begin{proof}
    T\"umevar\i mdan
    istedi\u gimiz \"ozellikleri olan bir g\"onderme varsa,
    tek bir \"ornek vard\i r.

    \c Simdi elemanlar\i\ g\"onderme olan
    bir $\mathscr C$ k\"umesini tan\i mlayaca\u g\i z.\label{curly-C}
    $\mathscr C$'nin her $h$ eleman\i\ i\c cin,
    \begin{compactenum}[1)]
    \item
      $h$'nin tan\i m k\"umesi 
      $\upomega$'nin bir altk\"umesidir, ve
    \item
      herhangi $\ell$ do\u gal say\i s\i\ i\c cin,
      $h(\ell)$ tan\i mlan\i rsa, o zaman
      \begin{compactenum}
      \item 
        ya $\ell=0$ ve $h(\ell)=b$,
      \item
        ya da bir $k$ do\u gal say\i s\i\ i\c cin $\ell=k+1$,
        $h(k)$ tan\i mlan\i r, ve
        \begin{equation*}
          h(\ell)=f(h(k)).
        \end{equation*}
      \end{compactenum}
    \end{compactenum}
    \.Istedi\u gimiz gibi $g$ g\"ondermesi varsa
    $\mathscr C$'nin eleman\i d\i r.
    Her $k$ do\u gal say\i s\i\ i\c cin,
    $A$'n\i n bir ve tek bir $d$ eleman\i\ i\c cin,
    $\mathscr C$'nin bir $h$ eleman\i\ i\c cin $h(k)=d$ g\"osterece\u giz.
    Bu \c sekilde $g(k)=d$ tan\i mlanabilir.

    Yukar\i daki \"ozelli\u gi olan $k$ do\u gal say\i lar\i,
    $E$ k\"umesini olu\c stursun.
    Tan\i m k\"umesi $\{0\}$ olan
    bir $h$ g\"ondermesi i\c cin $h(0)=b$.
    O zaman $h\in\mathscr C$.
    Ayr\i ca $\mathscr C$'nin herhangi $h$ eleman\i\ i\c cin $h(0)$ tan\i mlan\i rsa,
    o zaman $h(0)=b$ olmal\i d\i r, 
    \c c\"unk\"u hi\c c $k$ do\u gal say\i s\i\ i\c cin $k+1=0$ de\u gildir.
    Bu \c sekilde $0\in E$.

    \c Simdi $k\in E$ olsun.
    O zaman $A$'n\i n bir ve tek bir $d$ eleman\i\ i\c cin,
    $\mathscr C$'nin bir $h$ eleman\i\ i\c cin $h(k)=d$.
    \begin{compactenum}
    \item
      E\u ger $h(k+1)$ tan\i mlan\i rsa,
      o zaman $\mathscr C$'nin tan\i m\i na g\"ore
      $h(k+1)=f(d)$, 
      \c c\"unk\"u $k+1\neq0$, ve ayr\i ca herhangi $\ell$ do\u gal say\i s\i\ i\c cin
      e\u ger $\ell+1=k+1$ ise, o zaman $\ell=k$.
    \item
      E\u ger $h(k+1)$ tan\i mlanmazsa, 
      o zaman yeni bir $h^*$ g\"ondermesi i\c cin
      \begin{equation*}
        h^*(x)=\begin{cases}
	h(x),&\text{ e\u ger $h(x)$ tan\i mlan\i rsa,}\\
	f(d),&\text{ e\u ger $x=k+1$.}
        \end{cases}
      \end{equation*}
      O zaman $h^*\in\mathscr C$ ve $h^*(k+1)=f(d)$.
    \end{compactenum}
    Bu \c sekilde, her durumda,
    %$A$'n\i n $f(d)$ eleman\i\ i\c cin,
    $\mathscr C$'nin bir $h$ eleman\i\ i\c cin $h(k+1)=f(d)$.

    M\"umk\"umse $d^*\in A$, $d^*\neq f(d)$ olsun, 
    ama $\mathscr C$'nin bir $h$ eleman\i\ i\c cin $h(k+1)=d^*$ olsun.
    O zaman $k+1\neq0$ oldu\u gundan bir $\ell$ do\u gal say\i s\i\ i\c cin
    $\ell+1=k+1$, $h(\ell)$ tan\i mlan\i r, ve $d^*=f(h(\ell))$.
    Ama bu durumda $\ell=k$, dolay\i s\i yla $h(\ell)=d$ ve $d^*=f(d)$.

    Sonu\c c olarak $k+1\in E$.  T\"umevar\i m ile $E=\upomega$.
  \end{proof}

  \begin{sloppypar}
    Yukar\i daki kan\i t, sadece $\upomega$'nin a\c sa\u g\i daki \"ozelliklerini kullan\i r:
    \begin{compactenum}
    \item
      $0\in\upomega$.
    \item
      $k\in\upomega$ ise $k+1\in\upomega$.
    \item
      T\"umevar\i m y\"ontemi ge\c cerlidir.
    \item
      Her $k$ do\u gal say\i s\i\ i\c cin
      $0\neq k+1$.
    \item
      T\"um $k$ ve $\ell$ do\u gal say\i lar\i\ i\c cin
      $k+1=\ell+1$ ise $k=\ell$.
    \end{compactenum}
    Bu \"ozelliklere \textbf{Peano Aksiyomlar\i} denir.
    Peano Aksiyomlar\i,
    $\upomega$'da iki-konumlu toplama i\c sleminin tan\i mland\i\u g\i n\i\ varsaymaz;
    sadece tek-konumlu $x\mapsto x+1$ i\c slemi vard\i r.
    Ama \"ozyineleme y\"ontemiyle 
    $\upomega$'da toplama ve \c carpma i\c slemlerini tan\i mlayabiliriz:
    \begin{align*}
      a+(b+1)&=(a+b)+1,&a\cdot 1&=a,&a\cdot(b+1)&=ab+a.
    \end{align*}
    T\"umevar\i m ve kalan Peano Aksiyomlar\i\ ile
    toplaman\i n ve \c carpman\i n \"ozelliklerini kan\i tlayabiliriz;
    ayr\i ca $\upomega$'n\i n s\i ralamas\i n\i\ tan\i mlay\i p
    \"ozelliklerini kan\i tlayabiliriz.
    Ondan sonra
    B\"ol\"um \numarada{ch:intro}ki gibi
    $\N$, $\Z$, $\Q$ ve $\R$ yap\i lar\i n\i\ elde edebiliriz.
  \end{sloppypar}

  \begin{sloppypar}
    Do\u gal say\i lar, \emph{sonlu ordinallerdir.}
    Sonsuz ordinaller de vard\i r.
    Ordinallerin aksiyomlar\i n\i\ kullanarak
    toplama ve \c carpma i\c slemlerini tan\i mlayay\i p
    \"ozelliklerini kan\i tlayaca\u g\i z.
    Ondan sonra
    k\"ume aksiyomlar\i n\i\ kullanarak
    ordinalleri in\c sa edece\u giz.
    Bu \c sekilde bildi\u gimiz t\"um matematik,
    k\"ume aksiyomlar\i\ taraf\i ndan gerektirilir.
  \end{sloppypar}



\chapter{Ordinal Say\i lar}\label{ch:ord}

\section{Ordinaller}

E\u ger bir s\i n\i f\i n her eleman\i\ bir altk\"ume ise,
o zaman s\i n\i f \textbf{ge\c ci\c slidir.}
E\u ger $\bm A$ ge\c ci\c sli ise, o zaman
\begin{equation*}
  c\in b\land b\in\bm A\lto c\in\bm A.
\end{equation*}
\"Orne\u gin
\begin{compactitem}
  \item
    \Teoreme{thm:o-trans} g\"ore $\upomega$ ge\c ci\c slidir;
    \item
      \Teoreme{thm:n-trans} g\"ore
      $\upomega$'n\i n her eleman\i\ da ge\c ci\c slidir.
\end{compactitem}
Tan\i ma g\"ore
\begin{compactenum}[1)]
  \item
    ge\c ci\c sli olan
  \item
    $\in$ taraf\i ndan iyis\i ralanm\i\c s olan
\end{compactenum}
bir k\"ume bir \textbf{ordinal say\i d\i r.}
\begin{compactitem}
  \item
    \Teorem{thm:n-wo} sayes\i nde $\upomega$'n\i n her eleman\i\ bir ordinaldir;
    \item
  teoremin sonucu sayes\i nde $\upomega$ bir ordinaldir.
\end{compactitem}

\begin{xca}
  Bulun
  \begin{compactenum}[(a)]
    \item
  $\in$ ba\u g\i nt\i s\i n\i n ge\c ci\c sli oldu\u gu,
      ge\c ci\c sli olmayan bir k\"ume;
      \item
  $\in$ ba\u g\i nt\i s\i n\i n ge\c ci\c sli olmad\i\u g\i,
  ge\c ci\c sli olan bir k\"ume.
  \end{compactenum}
\end{xca}

Ordinaller
\begin{equation*}
  \on
\end{equation*}
s\i n\i f\i n\i\ olu\c stururlar.
$\on$'nin elemanlar\i\ her zaman
k\"u\c c\"uk Yunan harfleri taraf\i ndan g\"osterilecektir.
\"Ozel olarak $\alpha$, $\beta$, $\gamma$, $\delta$,
ve $\theta$,\label{minus-gr}
sabit ordinaldirler,
ama $\xi$, $\eta$, ve $\zeta$, ordinal de\u gi\c skendirler.
\"Ornegin
\begin{equation*}
\{\xi\colon\phi(\xi)\}
%=\{\eta\colon\phi(\eta)\}
%=\{\zeta\colon\phi(\zeta)\}
=\{x\colon x\in\on\land\phi(x)\}.
\end{equation*}

\begin{theorem}\label{thm:on-trans}
  $\on$ ge\c ci\c slidir,
  dolay\i s\i yla
 her ordinalin her eleman\i\ bir ordinaldir.
\end{theorem}

\begin{proof}
$\alpha\in\on$ be $b\in\alpha$ olsun.
O zaman $b\included\alpha$, dolay\i s\i yla
$\alpha$ gibi $b$,
$\in$ taraf\i ndan iyis\i ralanm\i\c st\i r.

\c Simdi $c\in b$ olsun.
O zaman $c\in\alpha$, dolay\i s\i yla $c\included\alpha$.
\"Ozel olarak $d\in c$ ise $d\in\alpha$.
Bu durumda $d$, $c$, ve $b$, $\alpha$'n\i n eleman\i d\i rlar;
ayr\i ca $d\in c$ ve $c\in b$,
dolay\i s\i yla $d\in b$ \c c\"unk\"u $\alpha$'da $\in$ ge\c ci\c slidir.
Sonu\c c olarak $c\included b$.
O halde $b$ ge\c ci\c slidir.
\end{proof}

\begin{lemma}\label{lem:on-o}
$\on$, $\in$ taraf\i ndan s\i ralanm\i\c st\i r.
\end{lemma}

\begin{proof}
$\alpha\in\on$ olsun.  $\alpha$'da $\in$ ba\u g\i nt\i s\i\ yans\i mas\i z oldu\u gundan
$\alpha\notin\alpha$,
\c c\"unk\"u $\alpha\in\alpha$ ise $\alpha$'n\i n bir $\beta$ eleman\i\ i\c cin $\beta\in\beta$.

E\u ger $\beta\in\alpha$ ve $\gamma\in\beta$ ise, 
$\alpha$ ge\c ci\c sli oldu\u gundan $\gamma\in\alpha$.
\end{proof}

\begin{lemma}
$\on$'de $\in$ ve $\pincluded$ s\i ralamalar\i\ ayn\i d\i r.
\end{lemma}

\begin{proof}
Kan\i t\i n iki par\c cas\i\ vard\i r.
\begin{asparaenum}
  \item
\fbox{\mathversion{bold}$\alpha\in\beta\lto\alpha\pincluded\beta$:}
$\alpha\in\beta$ olsun.
$\beta$ ge\c ci\c sli oldu\u gundan $\alpha\included\beta$.
$\beta$'da $\in$ yans\i mas\i z oldu\u gundan $\alpha\neq\beta$.
Bu \c sekilde $\alpha\pincluded\beta$.
\item
\fbox{\mathversion{bold}$\alpha\pincluded\beta\lto\alpha\in\beta$:}
$\alpha\pincluded\beta$ olsun.
O zaman $\beta\setminus\alpha$ k\"umesi bo\c s de\u gildir.
$\gamma=\min(\beta\setminus\alpha)$ olsun.
O zaman $\gamma\in\beta$.
Biz \fbox{\mathversion{bold}$\gamma=\alpha$} kan\i tlayaca\u g\i z.
Bu kan\i t\i n iki par\c cas\i\ vard\i r.
\begin{compactenum}
  \item
\fbox{\mathversion{bold}$\gamma\included\alpha$:}
$\delta\in\gamma$ olsun.
O zaman $\beta$ ge\c ci\c sli oldu\u gundan $\delta\in\beta$.
Ayr\i ca $\delta\notin\beta\setminus\alpha$, \c c\"unk\"u $\delta\in\min(\beta\setminus\alpha)$.
O halde $\delta\in\alpha$.
B\"oylece $\gamma\included\alpha$.
\item
\fbox{\mathversion{bold}$\alpha\included\gamma$:}
$\delta\in\alpha$ olsun.
O zaman $\delta\in\beta$, \c c\"unk\"u $\alpha\pincluded\beta$,
dolay\i s\i yla $\delta\notin\beta\setminus\alpha$.
Ama $\delta\in\gamma$, $\delta=\gamma$, veya $\gamma\in\delta$;
ve son iki imk\^an olmaz.
Zira $\gamma\in\beta\setminus\alpha$ oldu\u gundan $\delta\neq\gamma$;
ve $\gamma\notin\alpha$ oldu\u gundan $\gamma\notin\delta$, \c c\"unk\"u $\delta\in\alpha$.
Bu \c sekilde $\alpha\included\gamma$.\qedhere
\end{compactenum}
\end{asparaenum}
\end{proof}

\begin{theorem}
Her ordinalde $\in$ ve $\pincluded$ s\i ralamalar\i\ ayn\i d\i r.  
\end{theorem}

\ktk

\c Simdi $\on$'nin ve her eleman\i n\i n
$\in$ veya $\pincluded$ s\i ralamas\i n\i\ $<$ olarak yazabiliriz.

\begin{lemma}
$\on$'nin $<$ s\i ralamas\i\ do\u grusald\i r.
\end{lemma}

\begin{proof}
$\alpha\not\leq\beta$ olsun.
\fbox{\mathversion{bold}$\beta<\alpha$} g\"osterece\u giz.

Varsay\i mdan $\alpha\not\included\beta$,
dolay\i s\i yla $\alpha\setminus\beta\neq\emptyset$.
$\gamma=\min(\alpha\setminus\beta)$ olsun.
O zaman $\gamma\in\alpha$, yani $\gamma<\alpha$.
\fbox{\mathversion{bold}$\gamma=\beta$} g\"osterece\u giz.

\fbox{\mathversion{bold}$\gamma\included\beta$:}
$\delta\in\gamma$ olsun.
O zaman $\delta<\min(\alpha\setminus\beta)$,
ama $\delta\in\alpha$,
dolay\i s\i yla $\delta\in\beta$.

\fbox{\mathversion{bold}$\gamma\not\pincluded\beta$:}
$\gamma\in\alpha\setminus\beta$ oldu\u gundan $\gamma\notin\beta$,
yani $\gamma\not\pincluded\beta$.
\end{proof}

\begin{theorem}\label{thm:on-wo}
$\on$'nin $<$ do\u grusal s\i ralamas\i\ bir iyis\i ralamad\i r.
Asl\i nda $\on$'nin bo\c s olmayan her \emph{alts\i n\i f\i n\i n} en k\"u\c c\"uk eleman\i\ vard\i r.
\end{theorem}

\begin{proof}
$\bm A\included\on$ ve $\alpha\in\bm A$ olsun.
\begin{compactitem}
\item
$\alpha\cap\bm A=\emptyset$ ise
$\alpha=\min(\bm A)$.
\item
$\alpha\cap\bm A\neq\emptyset$ ise
$\min(\alpha\cap\bm A)=\min(\bm A)$.\qedhere
\end{compactitem}
\end{proof}

\begin{theorem}[Burali-Forti Paradoksu]\label{thm:BFP}
$\on$ k\"ume de\u gildir.
\end{theorem}

\begin{proof}
  \c Simdi Teorem \ref{thm:on-trans} ve \numarada{thm:on-wo}n
  $\on$ hem ge\c ci\c sli hem $\in$ taraf\i ndan iyis\i ralanm\i\c st\i r.
  Tan\i ma g\"ore $\on$'nin elemanlar\i n\i n ayn\i\ \"ozellikleri vard\i r.
  Ama $\on$ $\in$ taraf\i ndan s\i ralanm\i\c s oldu\u gundan kendinin eleman\i\ olamaz.
  Bu \c sekilde $\on$ k\"ume olamaz.
\end{proof}

\begin{theorem}\mbox{}
  \begin{compactenum}
    \item
      $\emptyset\in\on$.
    \item
      $\alpha\in\on$ ise $\alpha'\in\on$
ve ayr\i ca her $\beta$ ordinali i\c cin
\begin{equation*}
\beta\leq\alpha\lor\alpha'\included\beta.
\end{equation*}
  \end{compactenum}
\end{theorem}

\ktk

Tan\i ma g\"ore
ne $0$ ne bir ard\i l olan bir ordinal bir \textbf{limittir.}
O zaman $\upomega$ bir limittir
ve (\Teorem{thm:0-n'} sayesinde) en k\"u\c c\"uk limittir.
Sonsuzluk Aksiyomu'nu kullanmadan $\upomega$,
ne limit olan ne limit i\c ceren ordinallerin olu\c sturdu\u gu s\i n\i f
olarak tan\i mlanabilir.


\begin{theorem}\label{thm:'}
S\i f\i r olmayan bir $\alpha$ ordinalinin limit olmas\i\ i\c cin 
gerek ve yeter ko\c sul,
\begin{equation*}
\beta<\alpha\lto\beta'<\alpha.
\end{equation*}
\end{theorem}

\ktk


\begin{axiom}[Bile\c sim]
Her k\"umenin bile\c simi bir k\"umedir.
\end{axiom}

\begin{theorem}
  $\on$'n\i n her altk\"umesinin en k\"u\c c\"uk \"usts\i n\i r\i\ vard\i r.
  Asl\i nda
$B\pincluded\on$ ise
\begin{equation*}
  \sup B=\bigcup B.
\end{equation*}
\end{theorem}

\ktk


\begin{sloppypar}
  \c Simdi Burali-Forti Paradoksu'nun
  (yani \Teoremin{thm:BFP})
  ba\c ska bir kan\i t\i\ vard\i r.
  Her ordinalin daha b\"uy\"u\u g\"u oldu\u gundan
  $\on$'nin en b\"uy\"uk eleman\i\ yoktur,
  dolay\i s\i yla
  $\on$'nin $\on$'de olan \"usts\i n\i r\i\ yoktur.
  $\on$'nin her altk\"umesinin \"usts\i n\i r\i\ oldu\u gundan
  $\on$'nin kendisi k\"ume olamaz.
\end{sloppypar}



\section{T\"umevar\i m ve \"Ozyineleme}


\begin{theorem}[Ordinal T\"umevar\i m]
$\bm A\included\on$ olsun.
E\u ger
\begin{compactenum}[1)]
\item
$0\in\bm A$,
\item
Her $\beta$ i\c cin
\begin{equation*}
\beta\in\bm A\lto\beta'\in\bm A,
\end{equation*}
\item
her $\gamma$ limiti i\c cin
\begin{equation}\label{eqn:sub=>in}
\gamma\included\bm A\lto\gamma\in\bm A
\end{equation}
\end{compactenum}
ise, o zaman $\bm A=\on$.
\end{theorem}

\begin{proof}
Verilen ko\c sullar alt\i nda
$\on\setminus\bm A$ fark\i n\i n en k\"u\c c\"uk eleman\i\ olamaz.
Zira m\"umk\"umse $\alpha=\min(\on\setminus\bm A)$ olsun.
\begin{asparaenum}
\item
$\alpha=0$ ise $\alpha\in\bm A$.
\item
$\alpha=\beta'$ ise $\beta<\alpha$ oldu\u gundan $\beta\in\bm A$,
ama bu durumda $\beta'\in\bm A$, yani $\alpha\in\bm A$.
\item
Varsay\i m\i m\i za g\"ore
$\beta<\alpha$ ise $\beta\in\bm A$.
Bu \c sekilde
$\alpha\included\bm A$.
E\u ger ayr\i ca $\alpha$ bir limit ise,
o zaman \eqref{eqn:sub=>in} sayesinde
$\alpha$ da $\bm A$'n\i n eleman\i\ olmal\i d\i r.
\end{asparaenum}

Bu \c sekilde her ordinal ya $0$, ya bir ard\i l, ya da bir limit oldu\u gundan
$\alpha\in\bm A$,
ama $\alpha=\min(\on\setminus\bm A)$ varsay\i m\i na g\"ore $\alpha\notin\bm A$.
\"Oyleyse varsay\i m imk\^ans\i zd\i r.
$\on$'nin her bo\c s olmayan altk\"umesinin en k\"u\c c\"uk eleman\i\ var oldu\u gundan
$\on\setminus\bm A=\emptyset$,
dolay\i s\i yla $\bm A=\on$.
\end{proof}

Ordinal t\"umevar\i m ile
\Teoremi{thm:ord-rec}, 
\Teoremi{thm:norm-cond},
\Teoremi{thm:0+},
ve daha sonraki teoremler 
kan\i tlayaca\u g\i z.
Ordinal t\"umevar\i m kullan\i lan bir kan\i t\i n \"u\c c ad\i m\i\ vard\i r:
\begin{compactenum}[1)]
\item
s\i f\i r ad\i m\i,
\item
ard\i l ad\i m\i, ve
\item
limit ad\i m\i.
\end{compactenum}
Ayr\i ca kan\i tta iki t\"umevar\i m hipotezi vard\i r.
Ordinal T\"umevar\i m Teoremini yazarken kulland\i\u g\i m\i z harflerde,
\begin{compactitem}
\item
ard\i l ad\i m\i n\i n hipotezi, $\beta\in\bm A$;
\item
limit ad\i m\i n\i n hipotezi, $\gamma\included\bm A$,
yani
\begin{equation*}
\Forall{\xi}(\xi<\gamma\lto\xi\in\bm A).
\end{equation*}
\end{compactitem}


\begin{theorem}[Ordinal \"Ozyineleme]\label{thm:ord-rec}
%$\theta$ bir ordinal, $\bm F\colon\on\to\on$, ve $\bm G\colon\pow{\on}\to\on$ olsun.
Varsay\i mlar\i m\i z,
\begin{compactenum}[1)]
\item
$\theta\in\on$, 
\item
$\bm F\colon\on\to\on$.
%\item
%$\bm G\colon\pow{\on}\to\on$.
\end{compactenum}
O zaman bir ve tek bir $\bm H$ ordinal i\c slemi i\c cin
\begin{compactenum}[1)]
\item
$\bm H(0)=\theta$,
\item
her $\beta$ ordinali i\c cin $\bm H(\beta')=\bm F(\bm H(\beta))$,
\item
her $\gamma$ limiti i\c cin
$\bm H(\gamma)=\sup\{\bm H(\xi)\colon\xi<\gamma\}$.
\end{compactenum}
\end{theorem}

\begin{proof}
Her $\alpha$ i\c cin, 
tan\i m k\"umesi $\{\xi\colon\xi\leq\alpha\}$ olan
bir ve tek bir $h_{\alpha}$ g\"ondermesi i\c cin,
\begin{compactenum}[1)]
\item
$h_{\alpha}(0)=\theta$,
\item
$\beta<\alpha$ ise $h_{\alpha}(\beta')=\bm F(h_{\alpha}(\beta))$,
\item
$\gamma\leq\alpha$ ve limit ise $h_{\alpha}(\gamma)=\sup\{h_{\alpha}(\xi)\colon\xi<\gamma\}$.
\end{compactenum}
Bunu kan\i tlamak i\c cin, ordinal t\"umevar\i m kullanaca\u g\i z.
\begin{asparaenum}
\item
$h_0$, $h_0(0)=\theta$ ile tan\i mlanabilir ve tan\i mlanmal\i d\i r.
Yani $\alpha=0$ durumunda iddia do\u grudur.
\item
E\u ger $\alpha=\delta$ durumunda iddia do\u gru ise
$h_{\delta'}$,
\begin{equation*}
h_{\delta'}(\xi)=\begin{cases}
	h_{\delta}(\xi),&\text{ $\xi\leq\delta$ durumunda,}\\
	\bm F(h_{\delta}(\delta)),&\text{ $\xi=\delta'$ durumunda}
\end{cases}
\end{equation*}
kural\i\ taraf\i ndan tan\i mlanabilir.
Ayr\i ca $h_{\delta'}$ bu \c sekilde tan\i mlanmal\i d\i r,
\c c\"unk\"u hipoteze g\"ore
\begin{equation*}
h_{\delta'}\restriction\{\xi\colon\xi\leq\delta\}=h_{\delta}
\end{equation*}
olmal\i d\i r.
Bundan dolay\i\ $\alpha=\delta'$ durumunda iddia do\u grudur.
\item
Benzer \c sekilde bir $\delta$ i\c cin $\alpha<\delta$ durumlar\i nda iddia do\u gru ise,
o zaman $\alpha<\beta<\delta$ durumlar\i nda $h_{\alpha}(\alpha)=h_{\beta}(\alpha)$.
E\u ger ayr\i ca $\delta$ bir limit ise,
o zaman $h_{\delta}$,
\begin{equation*}
h_{\delta}(\xi)=\begin{cases}
	h_{\xi}(\xi),&\text{ $\xi<\delta$ durumunda,}\\
	\sup\{h_{\xi}(\xi)\colon\xi<\delta\},&\text{ $\xi=\delta$ durumunda}
\end{cases}
\end{equation*}
kural\i\ taraf\i ndan tan\i mlanabilir ve tan\i mlanmal\i d\i r,
ve bu \c sekilde $\alpha=\delta$ durumunda iddia do\u grudur.
\end{asparaenum}

Ordinal t\"umevar\i m\i m\i z bitti.
\c Simdi $\bm H(\xi)=h_{\xi}(\xi)$ tan\i mlanabilir ve tan\i mlanmal\i d\i r.
\end{proof}

B\"ol\"umler \ref{ch:add}, \ref{ch:mul}, ve \numarada{ch:exp}
ordinal \"ozyinelemeyle ordinal toplama, 
\c carpma, ve kuvvet alma i\c slemlerini tan\i mlayaca\u g\i z.

\section{Normal i\c slemler}

\c Simdi $\bm F$,
herhangi tek-konumlu ordinal i\c slem olsun.  
Ordinal aksiyomar\i na g\"ore $\{\bm F(\xi)\colon\xi<\alpha\}$ s\i n\i f\i\
her zaman bir k\"umedir,
ve bu k\"umenin \"usts\i n\i r\i\ vard\i r.
Ayr\i ca ordinaller iyis\i ralanm\i\c s oldu\u gundan
$\{\bm F(\xi)\colon\xi<\alpha\}$
k\"umesinin \"usts\i n\i rlar\i n\i n en k\"u\c c\"u\u g\"u vard\i r,
yani k\"umenin \textbf{supremumu} vard\i r.
Bu supremum, 
\begin{align*}
	&\sup\{\bm F(\xi)\colon\xi<\alpha\},&
	\sup_{\xi<\alpha}\bm F(\xi)
\end{align*}
\c sekillerinde yaz\i labilir.

\begin{theorem}\label{thm:sup-a}
  Her $\alpha$ ordinali i\c cin
  \begin{equation*}
    \sup\{\xi\colon\xi<\alpha\}=
    \begin{cases}
      0,&\text{ $\alpha=0$ durumunda},\\
\beta,&\text{ $\alpha=\beta'$ durumunda},\\
\alpha,&\text{ $\alpha$'n\i n limit oldu\u gu durumda.}
    \end{cases}
  \end{equation*}
\end{theorem}

\ktk

\begin{xca}
  $\{\xi'\colon\xi<\alpha\}$ k\"umesinin supremumunu hesaplay\i n.
\end{xca}

E\u ger
\begin{equation*}
\alpha\leq\beta\lto\bm F(\alpha)\leq\bm F(\beta)
\end{equation*}
ise, o zaman $\bm F$ \textbf{artand\i r.}
E\u ger
\begin{equation}\label{eqn:a+x}
\alpha<\beta\lto\bm F(\alpha)<\bm F(\beta)
\end{equation}
ise, o zaman $\bm F$ \textbf{kesin artand\i r.}
E\u ger
\begin{compactenum}[1)]
\item
$\bm F$ kesin artan ve 
\item
her $\alpha$ limiti i\c cin
$\bm F(\alpha)=\sup\{\bm F(\xi)\colon\xi<\alpha\}$
\end{compactenum}
ise, 
o zaman $\bm F$'ye \textbf{normal} densin.

\begin{xca}\label{ex:xx'}
$\xi\mapsto\xi'$ i\c sleminin kesin artan olup
normal olmad\i\u g\i n\i\ g\"osterin.
\end{xca}

\begin{xca}
Normal olan bir i\c slem \"orne\u gi verin.
\end{xca}

Sonraki teoremin ilk kullan\i m\i,
\Teoremin{thm:a+x} kan\i t\i nda olacakt\i r.

\begin{theorem}\label{thm:norm-cond}
$\bm F\colon\on\to\on$ olsun.
E\u ger
\begin{compactenum}[1)]
\item
her $\alpha$ i\c cin
$\bm F(\alpha)<\bm F(\alpha')$ ve 
\item
her $\alpha$ limiti i\c cin
$\bm F(\alpha)=\sup\{\bm F(\xi)\colon\xi<\alpha\}$
\end{compactenum}
ise, 
o zaman $\bm F$ normaldir.
\end{theorem}

\begin{proof}
$\bm F$'nin kesin artan oldu\u gunu g\"ostermek yeter.
\eqref{eqn:a+x} gerektirmesini
$\beta$ \"uzerinden t\"umevar\i m kullanarak
kan\i tlayaca\u g\i z.
  \begin{asparaenum}
    \item
    $\beta=0$ ise, \eqref{eqn:a+x} iddias\i\ do\u grudur, 
\c c\"unk\"u hi\c cbir zaman $\alpha<0$ de\u gildir.  
\item
$\beta=\gamma$ durumda \eqref{eqn:a+x} 
iddia do\u gru olsun.
E\u ger $\alpha<\gamma'$ ise, o zaman $\alpha\leq\gamma$, dolay\i s\i yla
\begin{align*}
      \bm F(\alpha)
&\leq\bm F(\gamma)&&\text{[t\"umevar\i m hipotezi]}\\
&<\bm F(\gamma').&&\text{[varsay\i m]}\\
\end{align*}
\item
$\gamma$ limit
ve $\alpha<\gamma$ ise, o zaman $\alpha<\alpha'<\gamma$, dolay\i s\i yla
\begin{align*}
  \bm F(\alpha)
&<\bm F(\alpha')&&\text{[varsay\i m]}\\
&\leq\sup\{\bm F(\xi)\colon\xi<\gamma\}&&\text{[supremumun tan\i m\i]}\\
&=\bm F(\gamma).&&\text{[varsay\i m]}
\end{align*}
(Bu ad\i mda bir t\"umevar\i m hipotezi kullan\i lm\i yor.)\qedhere
  \end{asparaenum}
\end{proof}

Sonraki teoremin ilk kullan\i m\i,
\Teoremin{thm:+assoc} kan\i t\i nda olacakt\i r.

\begin{theorem}\sloppy
$\bm F\colon\on\to\on$ ve normal olsun.
O zaman $\on$'nin bo\c s olmayan her $A$ altk\"umesi i\c cin
\begin{equation}\label{eqn:F(sup)}
\bm F(\sup(A))=\sup_{\xi\in A}\bm F(\xi).
\end{equation}
\end{theorem}

\begin{proof}
$A$ k\"umesinin supremumu $\alpha$ olsun.
$\bm F$ kesin artan oldu\u gundan
$\beta\in A$ ise $\bm F(\beta)\leq\bm F(\alpha)$.
Bundan dolay\i,
e\u ger $\alpha\in A$ ise, o zaman
\begin{equation*}
  \sup_{\xi\in A}\bm F(\xi)=F(\alpha),
\end{equation*}
yani \eqref{eqn:F(sup)} do\u grudur.
\c Simdi $\alpha\notin A$ olsun.
O zaman $\alpha$ ard\i l olamaz.
$A$ bo\c s olmad\i\u g\i ndan $\alpha=0$ olamaz,
dolay\i s\i yla $\alpha$ limittir.
Bu durumda $\bm F$ normal oldu\u gundan
\begin{equation}\label{eqn:F1}
  \bm F(\alpha)=\sup_{\xi<\alpha}\bm F(\xi).
\end{equation}
Ayr\i ca
\begin{equation}\label{eqn:F2}
\sup_{\xi\in A}\bm F(\xi)\leq\sup_{\xi<\alpha}\bm F(\xi),  
\end{equation}
\c c\"unk\"u $A\included\{\xi\colon\xi<\alpha\}$.
E\u ger $\beta<\alpha$ ise, 
$A$'n\i n bir $\gamma$ eleman\i\ i\c cin $\beta\leq\gamma<\alpha$,
dolay\i s\i yla $\bm F(\beta)\leq\bm F(\gamma)\leq\sup_{\xi\in A}\bm F(\xi)$.
Bu \c sekilde
\begin{equation}\label{eqn:F3}
  \sup_{\xi<\alpha}\bm F(\xi)\leq\sup_{\xi\in A}\bm F(\xi).
\end{equation}
Sonu\c c olarak
\eqref{eqn:F1}, \eqref{eqn:F2}, ve \eqref{eqn:F3} beraber
 \eqref{eqn:F(sup)} e\c sitli\u gini tekrar gerektirir.
\end{proof}

\section{S\"ureklilik}

Normallik kavram\i n\i n yerine
ger\c cel analizden gelen s\"ureklilik kavram\i n\i\ kullanabiliriz.
Ordinallerde,
kesin artan bir i\c slemin normal olmas\i\ i\c cin
gerek ve yeter bir ko\c sul,
i\c slemin s\"urekli olmas\i d\i r.
Bu sonu\c cu kurmak, bu altb\"ol\"um\"un i\c sidir.

Tekrar $\bm F\colon\on\to\on$ olsun.  Varsa,
$\bm F$'nin bir noktadaki s\"ureklili\u gi
ger\c cel analizdeki gibi tan\i mlan\i r.
Asl\i nda e\u ger
\begin{equation*}
\beta<\bm F(\alpha)<\gamma
\end{equation*}
ko\c sulunu sa\u glayan herhangi $\beta$ ve $\gamma$ i\c cin,
baz\i\ $\delta$ ve $\theta$ i\c cin,
\begin{equation*}
\delta<\alpha<\theta
\land\Forall{\xi}(\delta<\xi<\theta\lto\beta<\bm F(\xi)<\gamma)
\end{equation*}
ise, o zaman $\bm F$, $\alpha$'da \textbf{s\"ureklidir.}
E\u ger $\bm F(\alpha)=0$ veya $\alpha=0$ ise, 
o zaman $\beta=-1$ veya $\delta=-1$ olabilir.
Benzer \c sekilde \textbf{soldan} ve \textbf{sa\u gdan} olan s\"ureklilik
tan\i mlanabilir.

\begin{lemma}
$\on$'de her tek-konumlu i\c slem,
limit olmayan her noktada s\"ureklidir
ve her noktada sa\u gdan s\"ureklidir.
\end{lemma}

\klk

Ger\c cel analizdeki gibi
$\bm F\colon\on\to\on$ ise
\begin{equation*}
\limsup_{\xi\to\alpha^-}\bm F(\xi)=\min\left\{\sup_{\eta<\xi<\alpha}\bm F(\xi)\colon\eta<\alpha\right\}
\end{equation*}
tan\i m\i n\i\ yapar\i z.

\begin{lemma}
$\bm F\colon\on\to\on$ ve artan ise	
\begin{equation*}
\limsup_{\xi\to\alpha^-}\bm F(\xi)=\sup_{\xi<\alpha}\bm F(\xi).
\end{equation*}
\end{lemma}

\klk

\begin{lemma}
$\bm F\colon\on\to\on$ olsun.
$\bm F$ bir $\alpha$ limitinde s\"ureklidir ancak ve ancak
\begin{equation*}
\limsup_{\xi\to\alpha^-}\bm F(\xi)=\bm F(\alpha).
\end{equation*}
\end{lemma}

\klk

\begin{theorem}
$\bm F\colon\on\to\on$ ve kesin artan olsun.
O zaman $\bm F$ normaldir ancak ve ancak 
her noktada s\"ureklidir.
\end{theorem}

\ktk

\begin{xca}
  S\"urekli olup normal olmayan bir i\c slem \"orne\u gi verin.
\end{xca}

\chapter{Ordinal toplama}\label{ch:add}

\section{Tan\i m ve \"ozellikler}

\"Ozyineli tan\i ma g\"ore her $\alpha$ ordinali i\c cin
\begin{gather}\label{eqn:+0}
  \alpha+0=\alpha,\\\label{eqn:+'}
\alpha+\beta'=(\alpha+\beta)',\\\label{eqn:+lim}
\gamma\text{ limit ise }\alpha+\gamma=\sup\{\alpha+\xi\colon\xi<\gamma\}.
\end{gather}
Ordinal toplaman\i n \"ozelliklerinin \c co\u gu,
t\"umevar\i m ile kan\i tlan\i r;
ama ilk teoremimiz, t\"umevar\i mdan de\u gildir.

\begin{theorem}
  $\alpha+1=\alpha'$.
\end{theorem}

\begin{proof}
\hfill
$\begin{aligned}[t]
  \alpha+1
&=\alpha+0'&&\\
&=(\alpha+0)'&&\text{[\eqref{eqn:+'} tan\i m\i ndan]}\\
&=\alpha'.&&\text{[\eqref{eqn:+0} tan\i m\i ndan]}
\end{aligned}$\hfill\vspace{-0.8\baselineskip}

\mbox{}
\end{proof}

\begin{theorem}\label{thm:0+}
Her $\alpha$ i\c cin $0+\alpha=\alpha$.
\end{theorem}

\begin{proof}
Ordinal t\"umevar\i m kullanaca\u g\i z.
  \begin{asparaenum}
    \item
E\u ger $\alpha=0$ ise
\begin{align*}
0+\alpha
&=0+0&&\text{[varsay\i mdan]}\\
&=0&&\text{[\eqref{eqn:+0} tan\i m\i ndan]}\\
&=\alpha.&&\text{[varsay\i mdan]}
\end{align*}
\item
E\u ger
\begin{equation}\label{eqn:0+b=b}
0+\beta=\beta
\end{equation}
ise, o zaman
  \begin{align*}
    0+\beta'
&=(0+\beta)'&&\text{[\eqref{eqn:+'} tan\i m\i ndan]}\\
&=\beta'.&&\text{[\eqref{eqn:0+b=b} hipotezinden]}
  \end{align*}
\item
Bir $\alpha$ limiti i\c cin
\begin{equation}\label{eqn:0+b=b-lim}
  \Forall{\xi}(\xi<\alpha\lto0+\xi=\xi)
\end{equation}
ise, o zaman
\begin{align*}
  0+\alpha
&=\sup\{0+\xi\colon\xi<\alpha\}&&\text{[\eqref{eqn:+lim} tan\i m\i ndan]}\\
&=\sup\{\xi\colon\xi<\alpha\}&&\text{[\eqref{eqn:0+b=b-lim} hipotezinden]}\\
&=\alpha.&&\text{[\Teoremde{thm:sup-a}n]}\qedhere
\end{align*}
  \end{asparaenum}
\end{proof}

\begin{xca}
\mathversion{false}
A\c sa\u g\i daki kan\i t
nerede yanl\i\c st\i r?
\begin{falseproof}
Her $\alpha$ i\c cin $1+\alpha=\alpha'$ kan\i tlayaca\u g\i z.
\begin{asparaenum}
\item
$1+0=1=0'$.
\item
$1+\beta=\beta'$ ise, o zaman
\begin{equation*}
1+\beta'=(1+\beta)'=(\beta')'.
\end{equation*}
\item
$\gamma$ limit ve $\Forall{\xi}(\xi<\gamma\lto1+\xi=\xi')$ ise
\begin{equation*}
1+\gamma=\sup_{\xi<\gamma}(1+\xi)=\sup_{\xi<\gamma}(\xi')=\gamma'.
\end{equation*}
\end{asparaenum}

B\"oylece her $\alpha$ i\c cin $1+\alpha=\alpha'$.
\end{falseproof}
\end{xca}

\begin{theorem}\label{thm:a+x}
Her $\alpha$ ordinali i\c cin $\xi\mapsto\alpha+\xi$ normaldir.
\end{theorem}

\begin{proof}
\Teoremde{thm:norm-cond}n $\alpha+\beta<\alpha+\beta'$ g\"ostermek yeter.
Ayr\i ca
\begin{align*}
      \alpha+\beta
&<(\alpha+\beta)'&&\\
&=\alpha+\beta'.&&\text{[\eqref{eqn:+'} tan\i m\i ndan]}\qedhere
\end{align*}
\end{proof}

\"Orne\u gin
\Sekle{fig:w+x} bak\i n.
\begin{figure}
  \centering
\psset{unit=10.8mm,dotsize=6pt}
\begin{pspicture}(10.24,10.24)
\uput[d](0,0){$0$}
\uput[l](0,0){$0$}
\psline(0,0)(10.24,0)
\psline(0,0)(0,10.24)
\psset{linestyle=dotted}
\psline(! 0 5.12)(! 10.24 5.12)
\psline(! 0 5.12 2.56 add)(! 10.24 5.12 2.56 add)
\psline(! 0 5.12 2.56 add 1.28 add)(! 10.24 5.12 2.56 add 1.28 add)
\psline(! 0 5.12 2.56 add 1.28 add 0.64 add)(! 10.24 5.12 2.56 add 1.28 add 0.64 add)
\psline(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add)(! 10.24 5.12 2.56 add 1.28 add 0.64 add 0.32 add)
\psline(0,10.24)(10.24,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psline(! 5.12 0)(! 5.12 10.24)
\psline(! 5.12 2.56 add 0)(! 5.12 2.56 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0)(! 5.12 2.56 add 1.28 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0.64 add 0)(! 5.12 2.56 add 1.28 add 0.64 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0)(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 10.24)
\psline(10.24,0)(10.24,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Xi-axis
\psdots[dotstyle=|]
(2.56,0)
(! 2.56 1.28 add 0)
(! 2.56 1.28 add 0.64 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(5.12, 0)
\uput[ur](2.56,0){$1$}
\uput[ur](! 5.12 1.28 sub 0){$2$}
\uput[ur](! 5.12 0.64 sub 0){$3$}
\uput[ur](5.12,0){$\upomega$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 1.28 add 0)
(! 5.12 1.28 add 0.64 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 0)
\uput[u]{45}(! 5.12 1.28 add 0){$\upomega+1$}
\uput[u]{45}(! 10.24 2.56 sub 0){$\upomega\cdot2$}
\uput[u]{45}(! 10.24 1.28 sub 0){$\upomega\cdot3$}
\uput[u]{45}(10.24,0){$\upomega^2$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 0.64 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.32 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(10.24, 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Eta-axis
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(0,2.56)
(! 0 2.56 1.28 add)
(! 0 2.56 1.28 add 0.64 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(0,5.12)
\uput[ur](0,2.56){$1$}
\uput[ur](! 0 5.12 1.28 sub){$2$}
\uput[ur](! 0 5.12 0.64 sub){$3$}
\uput[ur](0,5.12){$\upomega$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 1.28 add)
(! 0 5.12 1.28 add 0.64 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add)
\uput[ur](! 0 5.12 1.28 add){$\upomega+1$}
\uput[ur](! 0 10.24 2.56 sub){$\upomega\cdot2$}
\uput[ur](! 0 10.24 1.28 sub){$\upomega\cdot3$}
\uput[ur](0,10.24){$\upomega^2$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 0.64 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.32 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(0,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots: 
\psdots
(0,5.12)
(! 2.56          5.12 1.28 add)
(! 5.12 1.28 sub 7.68 0.64 sub)
(! 5.12 0.64 sub 7.68 0.32 sub)
(! 5.12 0.32 sub 7.68 0.16 sub)
(! 5.12 0.16 sub 7.68 0.08 sub)
(! 5.12 0.08 sub 7.68 0.04 sub)
(! 5.12 0.04 sub 7.68 0.02 sub)
(! 5.12 0.02 sub 7.68 0.01 sub)
(5.12,7.68)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 1.28 add                                                       7.68 0.64 add)
(! 5.12 1.28 add 0.64 add                                              7.68 0.64 add 0.32 add)
(! 5.12 1.28 add 0.64 add 0.32 add                                     7.68 0.64 add 0.32 add 0.16 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add                            7.68 0.64 add 0.32 add 0.16 add 0.08 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add                   7.68 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add          7.68 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 7.68 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add                                                       7.68 1.28 add)
(! 5.12 2.56 add 0.64 add                                              7.68 1.28 add 0.32 add)
(! 5.12 2.56 add 0.64 add 0.32 add                                     7.68 1.28 add 0.32 add 0.16 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add                            7.68 1.28 add 0.32 add 0.16 add 0.08 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add                   7.68 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add          7.68 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 7.68 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add                                              7.68 1.28 add 0.64 add)
(! 5.12 2.56 add 1.28 add 0.32 add                                     7.68 1.28 add 0.64 add 0.16 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add                            7.68 1.28 add 0.64 add 0.16 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add                   7.68 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add          7.68 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 7.68 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add                                     7.68 1.28 add 0.64 add 0.32 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add                            7.68 1.28 add 0.64 add 0.32 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add                   7.68 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add          7.68 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 7.68 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add                            7.68 1.28 add 0.64 add 0.32 add 0.16 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add                   7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add          7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add                   7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add          7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add          7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots(! 10.24 0.02 sub 10.24 0.01 sub)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdot(10.24,10.24)
\end{pspicture}

  \caption{$\eta=\upomega+\xi$ denkleminin grafi\u gi}\label{fig:w+x}  
\end{figure}
\c Sekilde
\begin{align*}
	\upomega\cdot2&=\upomega+\upomega,&
\upomega\cdot3&=\upomega\cdot2+\upomega,&
\upomega\cdot4&=\upomega\cdot3+\upomega,
\end{align*}
ve genelde,
\Teoremi{thm:rec} kullanan resmi \"ozyineli tan\i ma g\"ore,
\begin{align*}
\alpha\cdot0&=0,&
	\alpha\cdot1&=\alpha,&
	\alpha\cdot(k+1)&=\alpha\cdot k+\alpha.
\end{align*}
Bu \c sekilde $\alpha\cdot n$, ``$\alpha$'d\i r $n$ kere''
veya ``$\alpha$'n\i n $n$ kat\i d\i r.''
Ayr\i ca
\begin{equation*}
\upomega^2=\upomega\cdot\upomega=\sup_{x<\upomega}(\upomega\cdot x).
\end{equation*}

\begin{xca}
$\xi\mapsto\xi\cdot2$ g\"ondermesi kesin artan m\i d\i r?
S\"urekli midir?
\end{xca}

\begin{xca}
\mathversion{false}
A\c sa\u g\i daki kan\i t
nerede yanl\i\c st\i r?
\begin{falseproof}
Her $\alpha$ i\c cin, her $\beta$ i\c cin,
$\alpha+\beta=\beta+\alpha$ kan\i tlayaca\u g\i z.
\begin{asparaenum}
\item
$\alpha+0=\alpha=0+\alpha$.
\item
E\u ger $\alpha+\beta=\beta+\alpha$ ise, o zaman
\begin{equation*}
\alpha+\beta'=(\alpha+\beta)'=(\beta+\alpha)'=\beta'+\alpha.
\end{equation*}
\item
E\u ger $\gamma$ limit ve $\Forall{\xi}(\xi<\gamma\lto\alpha+\xi=\xi+\alpha)$ ise,
o zaman
\begin{equation*}
\alpha+\gamma
=\sup_{\xi<\gamma}(\alpha+\xi)
=\sup_{\xi<\gamma}(\xi+\alpha)
=\gamma+\alpha.
\end{equation*}
\end{asparaenum}
Bu \c sekilde her durumda $\alpha+\beta=\beta+\alpha$.
\end{falseproof}
\end{xca}

\begin{theorem}\label{thm:+assoc}
Ordinal toplama birle\c smelidir.
\end{theorem}

\begin{proof}
Her $\gamma$ i\c cin, 
t\"umevar\i m kullanarak
t\"um $\alpha$ ve $\beta$ i\c cin
  \begin{equation*}
    \alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma
  \end{equation*}
	g\"osterece\u giz.
  \begin{asparaenum}
    \item
$		\begin{aligned}[t]
\alpha+(\beta+0)
&=\alpha+\beta&&\text{[\eqref{eqn:+0} tan\i m\i ndan]}\\
&=(\alpha+\beta)+0.&&\text{[\eqref{eqn:+0} tan\i m\i ndan]}
					\end{aligned}$
\item
E\u ger
\begin{equation}\label{eqn:+assoc'}
\alpha+(\beta+\delta)=(\alpha+\beta)+\delta
\end{equation}
ise, o zaman
  \begin{align*}
    \alpha+(\beta+\delta')
		&=\alpha+(\beta+\delta)'&&\text{[\eqref{eqn:+'} tan\i m\i ndan]}\\
&=(\alpha+(\beta+\delta))'&&\text{[\eqref{eqn:+'} tan\i m\i ndan]}\\
&=((\alpha+\beta)+\delta)'&&\text{[\eqref{eqn:+assoc'} hipotezinden]}\\
&=(\alpha+\beta)+\delta'.&&\text{[\eqref{eqn:+'} tan\i m\i ndan]}
  \end{align*}
\item
$\delta$ limit olsun, ve
\begin{equation}\label{eqn:+assoc-lim}
\Forall{\xi}\bigl(\xi<\delta\lto
\alpha+(\beta+\xi)=(\alpha+\beta)+\xi\bigr)
\end{equation}
olsun.  
O zaman
\begin{align*}
&\phantom{{}={}}(\alpha+\beta)+\delta\\
&=\sup\{(\alpha+\beta)+\xi\colon\xi<\delta\}&&\text{[\eqref{eqn:+lim} tan\i m\i]}\\
&=\sup\{\alpha+(\beta+\xi)\colon\xi<\delta\}&&\text{[\eqref{eqn:+assoc-lim} hipotezi]}\\
&=\alpha+\sup\{\beta+\xi\colon\xi<\delta\}&&\text{[$\xi\mapsto\alpha+\xi$ normaldir]}\\
&=\alpha+(\beta+\delta).&&\text{[\eqref{eqn:+lim} tan\i m\i]}\qedhere 
\end{align*}
  \end{asparaenum}
\end{proof}

\c Simdi herhangi $n$ sayma say\i s\i\ i\c cin
\begin{equation*}
\alpha\cdot n=\underbrace{\alpha+\dots+\alpha}_n
\end{equation*}
anla\c s\i labilir.

\begin{theorem}
$k<\upomega$ ve $\ell<\upomega$ ise
$\alpha\cdot(k+\ell)=\alpha\cdot k+\alpha\cdot\ell$.
\end{theorem}

\ktk[ (T\"umevar\i m kullan\i n.)]

\begin{theorem}\label{thm:x+a}
Her $\xi\mapsto\xi+\alpha$ g\"ondermesi artand\i r.
\end{theorem}

\begin{proof}
  $\beta\leq\gamma$ olsun.
$\alpha$ \"uzerinden t\"umevar\i m kullanarak  
\begin{equation*}
\beta+\alpha\leq\gamma+\alpha
\end{equation*}
kan\i tlayaca\u g\i z.
\begin{asparaenum}
  \item
$\beta+0=\beta\leq\gamma=\gamma+0$.
\item
$\beta+\alpha=\gamma+\alpha$ ise tabii ki
  \begin{equation*}
    \beta+\alpha'=(\beta+\alpha)'=(\gamma+\alpha)'=\gamma+\alpha'.
  \end{equation*}
$\beta+\alpha<\gamma+\alpha$ ise, 
\Teoreme{thm:'} g\"ore
\begin{equation*}
  \beta+\alpha'
  =(\beta+\alpha)'\leq\gamma+\alpha<(\gamma+\alpha)'=\gamma+\alpha'.  
\end{equation*}
\item
E\u ger $\delta$ limit ise
\begin{equation*}
\Forall{\xi}(\xi<\delta\lto\beta+\xi<\gamma+\xi)
\end{equation*}
olsun.  O zaman
  \begin{equation*}
    \beta+\delta=\sup_{\xi<\delta}(\beta+\xi)
\leq\sup_{\xi<\delta}(\gamma+\xi)=\gamma+\delta.\qedhere
  \end{equation*}
  \end{asparaenum}
\end{proof}

\section{Hesaplamalar}

Bu altb\"ol\"um\"un teoremleri
t\"umevar\i m kullanmaz.

\begin{theorem}%\label{thm:k+w=w}
  $k<\upomega$ ise $k+\upomega=\upomega$.
(\Sekle{fig:x+w} bak\i n.)
\begin{figure}
  \centering
\psset{unit=10.8mm,dotsize=6pt}
\begin{pspicture}(10.24,10.24)
\uput[d](0,0){$0$}
\uput[l](0,0){$0$}
\psline(0,0)(10.24,0)
\psline(0,0)(0,10.24)
\psset{linestyle=dotted}
\psline(! 0 5.12)(! 10.24 5.12)
\psline(! 0 5.12 2.56 add)(! 10.24 5.12 2.56 add)
\psline(! 0 5.12 2.56 add 1.28 add)(! 10.24 5.12 2.56 add 1.28 add)
\psline(! 0 5.12 2.56 add 1.28 add 0.64 add)(! 10.24 5.12 2.56 add 1.28 add 0.64 add)
\psline(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add)(! 10.24 5.12 2.56 add 1.28 add 0.64 add 0.32 add)
\psline(0,10.24)(10.24,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psline(! 5.12 0)(! 5.12 10.24)
\psline(! 5.12 2.56 add 0)(! 5.12 2.56 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0)(! 5.12 2.56 add 1.28 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0.64 add 0)(! 5.12 2.56 add 1.28 add 0.64 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0)(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 10.24)
\psline(10.24,0)(10.24,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Xi-axis
\psdots[dotstyle=|]
(2.56,0)
(! 2.56 1.28 add 0)
(! 2.56 1.28 add 0.64 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(5.12, 0)
\uput[ur](2.56,0){$1$}
\uput[ur](! 5.12 1.28 sub 0){$2$}
\uput[ur](! 5.12 0.64 sub 0){$3$}
\uput[ur](5.12,0){$\upomega$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 1.28 add 0)
(! 5.12 1.28 add 0.64 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 0)
\uput[u]{45}(! 5.12 1.28 add 0){$\upomega+1$}
\uput[u]{45}(! 10.24 2.56 sub 0){$\upomega\cdot2$}
\uput[u]{45}(! 10.24 1.28 sub 0){$\upomega\cdot3$}
\uput[u]{45}(10.24,0){$\upomega^2$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 0.64 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.32 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(10.24, 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Eta-axis
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(0,2.56)
(! 0 2.56 1.28 add)
(! 0 2.56 1.28 add 0.64 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(0,5.12)
\uput[ur](0,2.56){$1$}
\uput[ur](! 0 5.12 1.28 sub){$2$}
\uput[ur](! 0 5.12 0.64 sub){$3$}
\uput[ur](0,5.12){$\upomega$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 1.28 add)
(! 0 5.12 1.28 add 0.64 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add)
\uput[ur](! 0 5.12 1.28 add){$\upomega+1$}
\uput[ur](! 0 10.24 2.56 sub){$\upomega\cdot2$}
\uput[ur](! 0 10.24 1.28 sub){$\upomega\cdot3$}
\uput[ur](0,10.24){$\upomega^2$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 0.64 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.32 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(0,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots: 
\psdots
(0,5.12)
(! 2.56          5.12)
(! 5.12 1.28 sub 5.12)
(! 5.12 0.64 sub 5.12)
(! 5.12 0.32 sub 5.12)
(! 5.12 0.16 sub 5.12)
(! 5.12 0.08 sub 5.12)
(! 5.12 0.04 sub 5.12)
(! 5.12 0.02 sub 5.12)
\psdot[dotstyle=o](5.12,5.12)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(5.12,7.68)
(! 5.12 1.28 add                                                       7.68)
(! 5.12 1.28 add 0.64 add                                              7.68)
(! 5.12 1.28 add 0.64 add 0.32 add                                     7.68)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add                            7.68)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add                   7.68)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add          7.68)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 7.68)
\psdot[dotstyle=o]
(! 5.12 2.56 add                                                       7.68)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add                                                       7.68 1.28 add)
(! 5.12 2.56 add 0.64 add                                              7.68 1.28 add)
(! 5.12 2.56 add 0.64 add 0.32 add                                     7.68 1.28 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add                            7.68 1.28 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add                   7.68 1.28 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add          7.68 1.28 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 7.68 1.28 add)
\psdot[dotstyle=o]
(! 5.12 2.56 add 1.28 add                                              7.68 1.28 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add                                              7.68 1.28 add 0.64 add)
(! 5.12 2.56 add 1.28 add 0.32 add                                     7.68 1.28 add 0.64 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add                            7.68 1.28 add 0.64 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add                   7.68 1.28 add 0.64 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add          7.68 1.28 add 0.64 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 7.68 1.28 add 0.64 add)
\psdot[dotstyle=o]
(! 5.12 2.56 add 1.28 add 0.64 add                                     7.68 1.28 add 0.64 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add                                     7.68 1.28 add 0.64 add 0.32 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add                            7.68 1.28 add 0.64 add 0.32 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add                   7.68 1.28 add 0.64 add 0.32 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add          7.68 1.28 add 0.64 add 0.32 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 7.68 1.28 add 0.64 add 0.32 add)
\psdot[dotstyle=o]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add                            7.68 1.28 add 0.64 add 0.32 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add                            7.68 1.28 add 0.64 add 0.32 add 0.16 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add                   7.68 1.28 add 0.64 add 0.32 add 0.16 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add          7.68 1.28 add 0.64 add 0.32 add 0.16 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 7.68 1.28 add 0.64 add 0.32 add 0.16 add)
\psdot[dotstyle=o]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add                   7.68 1.28 add 0.64 add 0.32 add 0.16 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add                   7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add          7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
\psdot[dotstyle=o]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add          7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add          7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
\psdot[dotstyle=o]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 7.68 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots(! 10.24 0.02 sub 10.24 0.01 sub)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdot(10.24,10.24)
\end{pspicture}

  \caption{$\eta=\xi+\upomega$ denkleminin grafi\u gi}\label{fig:x+w}  
\end{figure}
\end{theorem}

\begin{proof}
  $k+\upomega=\sup\{k+x\colon x<\upomega\}=\upomega$.
\end{proof}

\begin{corollary}
$k<\upomega$ ve $1\leq n<\upomega$ ise
\begin{equation*}
k+\upomega\cdot n=\upomega\cdot n.
\end{equation*}
\end{corollary}

\ksk

\begin{comment}

\begin{corollary}
  Ordinal toplama de\u gi\c smeli de\u gildir.
\end{corollary}

\ksk
\end{comment}

\begin{theorem}[\c C\i karma]\label{thm:subtraction}
  $\alpha\leq\beta$ ise
  \begin{equation}\label{eqn:a+x=b}
    \alpha+\xi=\beta
  \end{equation}
denkleminin bir ve tek bir \c c\"oz\"um\"u vard\i r.  
\end{theorem}

\begin{proof}
Denklemin \c c\"oz\"um\"u varsa,
\Teoreme{thm:a+x} g\"ore tek \c c\"oz\"um vard\i r.

Teoremler \ref{thm:0+} ve \numarada{thm:x+a}n $\alpha+\beta\geq\beta$,
dolay\i s\i yla $\{\xi\colon\alpha+\xi\leq\beta\}$ 
s\i n\i f\i n\i n $\beta'$ \"usts\i n\i r\i\ vard\i r.
\c Simdi $\gamma$, s\i n\i f\i n\i n supremumu olsun.
O zaman
\begin{gather*}
\begin{aligned}
	\alpha+\gamma
	&=\alpha+\sup\{\xi\colon\alpha+\xi\leq\beta\}\\
	&=\sup\{\alpha+\xi\colon\alpha+\xi\leq\beta\}
	\leq\beta,
\end{aligned}\\
(\alpha+\gamma)'
		=\alpha+\gamma'
		>\beta,
\end{gather*}
dolay\i s\i yla $\alpha+\gamma=\beta$.
\end{proof}

\begin{xca}
$\alpha\leq\beta$ varsay\i nca,
$\{\xi\colon a+\xi\geq\beta\}$ s\i n\i f\i n\i n bo\c s olmay\i p
s\i n\i f\i n en k\"u\c c\"uk eleman\i n\i n \eqref{eqn:a+x=b} denkleminin
\c c\"oz\"um\"u oldu\u gunu g\"osterin.
\end{xca}

\begin{theorem}
$\upomega+\alpha=\alpha$ ancak ve ancak $\upomega^2\leq\alpha$.
\end{theorem}

\begin{proof}
$\begin{aligned}[t]
	\upomega+\upomega^2
	&=\upomega+\sup_{x<\upomega}(\upomega\cdot x)\\
	&=\sup_{x<\upomega}(\upomega+\upomega\cdot x)\\
	&=\sup_{x<\upomega}\bigl(\upomega\cdot(1+x)\bigr)\\
	&=\upomega^2.
\end{aligned}$

E\u ger $\alpha\geq\upomega^2$ ise, o zaman bir $\beta$ i\c cin $\upomega^2+\beta=\alpha$,
dolay\i s\i yla
\begin{equation*}
\upomega+\alpha
=\upomega+\upomega^2+\beta
=\upomega^2+\beta
=\alpha.
\end{equation*}
\c Simdi $\alpha<\upomega^2$ olsun.
O zaman bir $k$ do\u gal say\i s\i\ i\c cin
\begin{gather*}%\label{eqn:wk-a-w(k+1)}
\upomega\cdot k\leq\alpha<\upomega\cdot(k+1),\\%\notag
\upomega\cdot(k+1)\leq\upomega+\alpha,	
\end{gather*}
dolay\i s\i yla $\alpha<\upomega+\alpha$.
\end{proof}

Teorem sayesinde $\upomega\leq\alpha<\upomega^2$ ise,
o zaman bir $\alpha_1$ i\c cin
\begin{align*}
	\upomega+\alpha_1&=\alpha,&
	\alpha_1&<\alpha.
\end{align*}
E\u ger $\upomega\leq\alpha_1$ ise,
o zaman bir $\alpha_2$ i\c cin
\begin{align*}
	\upomega+\alpha_2&=\alpha_1,&
	\alpha_2&<\alpha_1,
\end{align*}
ve saire.
O zaman bir $k$ i\c cin
\begin{equation*}
\alpha
={\underbrace{\upomega+\dots+\upomega}_k}+\alpha_k
=\upomega\cdot k+\alpha_k.
\end{equation*}
$\on$ iyis\i ral\i\ oldu\u gundan bir $k$ i\c cin
$\alpha_k<\upomega$.
Bu \c sekilde
\begin{equation*}
\{\xi\colon\xi<\upomega^2\}
\end{equation*}
k\"umesinin her eleman\i,
$\upomega\cdot k+\ell$ bi\c ciminde yaz\i labilir.
Verilen k\"ume, toplama alt\i nda kapal\i d\i r,
ve toplama kural\i,
\begin{equation*}
(\upomega\cdot k+\ell)+(\upomega\cdot m+n)
=\upomega\cdot(k+m)+n.
\end{equation*}

\begin{xca}
$\alpha=\upomega\cdot17+6$ ve $\beta=\upomega\cdot1000+5$ ise
$\alpha+\beta$ toplam\i n\i\ hesaplay\i n.
\end{xca}

\section{Kardinaller}

\c Simdi herhangi $A$ ve $B$ k\"umeleri i\c cin
\begin{equation*}
  A\sqcup B=(A\times\{0\})\cup(B\times\{1\})
\end{equation*}
olsun; bu bile\c sim, 
$A$ ve $B$'nin \textbf{ayr\i k bile\c simidir.}
B\"ol\"um \numarada{ch:ax}n
\begin{equation*}
\alpha=\{\xi\colon\xi<\alpha\}
\end{equation*}
anla\c smas\i n\i\ kullanaca\u g\i z.

\begin{theorem}\label{thm:a+b}
  $\alpha+\beta\approx\alpha\sqcup\beta$.
\end{theorem}

\begin{proof}
  \Teoremde{thm:subtraction}n
	\begin{equation*}
	\left\{
	\begin{aligned}
(\xi,0)&\mapsto\xi,\\
(\eta,1)&\mapsto\alpha+\eta
	\end{aligned}
	\right.
	\end{equation*}
kural\i,
$\alpha\sqcup\beta$
ayr\i k bile\c siminden
$\alpha+\beta$ k\"umesine
giden bir e\c sleme tan\i mlar.
\end{proof}

Bir $A$ k\"umesi bir ordinalle e\c slenik olsun.
Tan\i ma g\"ore
\begin{equation*}
\card A=\min\bigl\{\xi\colon\xi\approx A\bigr\};
\end{equation*}
bu ordinal, $A$'n\i n \textbf{kardinalidir.}
Kardinaller, $\kappa$, $\lambda$, $\mu$, ve $\nu$\label{card}
harfleri ile g\"osterilecektir.

E\u ger $f\colon A\to B$ ve $C\included A$ ise
\begin{equation*}
f[C]=\{f(x)\colon x\in C\}
\end{equation*}
olsun.
E\u ger $f$ birebir ise, o zaman
$A$'n\i n $B$'ye bir \textbf{g\"ommesidir,}
ve
\begin{equation*}
A\approx f[A]\included B.
\end{equation*}
Bu durumda
\begin{equation*}
f\colon A\xrightarrow{\preccurlyeq}B
\end{equation*}
yazal\i m, ve \"oyle bir $f$ g\"ommesi varsa
\begin{equation*}
A\preccurlyeq B
\end{equation*}
yazal\i m.

\begin{theorem}[Schr\"oder--Bernstein]
$A\preccurlyeq B$ ve $B\preccurlyeq A$ ise
\begin{equation*}
A\approx B.
\end{equation*}
\end{theorem}

\begin{proof}
$f\colon A\xrightarrow{\preccurlyeq}B$ ve
$g\colon B\xrightarrow{\preccurlyeq}A$ olsun.
\"Ozyinelemeyle
\begin{align*}
A_0&=A,&A_{n+1}&=g[B_n],\\
B_0&=B,&B_{n+1}&=f[A_n]
\end{align*}
olsun.
O zaman
\begin{gather*}
	f[A_0\setminus A_1]=B_1\setminus B_2,\\
	g[B_0\setminus B_1]=A_1\setminus A_2,
\end{gather*}
dolay\i s\i yla $A_0\setminus A_2\approx B_0\setminus B_2$.
Benzer \c sekilde
\begin{equation*}
A_n\setminus A_{n+2}\approx B_n\setminus B_{n+2},
\end{equation*}
dolay\i s\i yla
\begin{equation*}
A\setminus\bigcap_{i<\upomega}A_i\approx B\setminus\bigcap_{i<\upomega}B_i.
\end{equation*}
Ayr\i ca
\begin{equation*}
f\left[\bigcap_{i<\upomega}A_i\right]=\bigcap_{i<\upomega}f[A_i]=\bigcap_{0<i<\upomega}B_i=\bigcap_{i<\upomega}B_i,
\end{equation*}
dolay\i s\i yla 
$\bigcap_{i<\upomega}A_i\approx\bigcap_{i<\upomega}B_i$,
ve sonu\c c olarak $A\approx B$.
\end{proof}

\begin{theorem}
$\xi\mapsto\card{\xi}$ artand\i r.
\end{theorem}

\begin{proof}
E\u ger $\alpha\leq\beta$ ama $\card{\beta}\leq\card{\alpha}$ ise,
o zaman
\begin{equation*}
\alpha\preccurlyeq\beta\approx\card{\beta}\preccurlyeq\card{\alpha}\approx\alpha,
\end{equation*}
dolay\i s\i yla $\alpha\approx\beta$.
\end{proof}

\begin{theorem}
$k<\upomega$ ise $\card k=k$.
\end{theorem}

\ktk

\begin{theorem}
$\card{\upomega+\upomega}=\upomega$.
\end{theorem}

\ktk

\begin{corollary}
$\{\xi\colon\upomega\leq\xi<\upomega^2\}$ k\"umesinin her eleman\i n\i n kardinali
$\upomega$'d\i r.
\end{corollary}

\ksk

\chapter{Ordinal \c carpma}\label{ch:mul}

\section{Tan\i m ve \"ozellikler}

\"Ozyineli tan\i ma g\"ore her $\alpha$ i\c cin
\begin{gather*}
	\alpha\cdot0=0,\\
	\alpha\cdot\beta'=\alpha\cdot\beta+\alpha,\\
	\gamma\text{ limit ise }\alpha\cdot\gamma=\sup\{\alpha\cdot\xi\colon\xi<\gamma\}.
\end{gather*}

Ordinal \c carpma hakk\i nda ilk teoremimizin bir \c s\i kk\i\ t\"umevar\i m kullanmaz;
kalanlar t\"umevar\i m kullan\i yor.

\begin{theorem}\mbox{}
\begin{compactenum}
\item
$\alpha\cdot1=\alpha$.
\item
$1\cdot\alpha=\alpha$.
\item
$0\cdot\alpha=0$.
\end{compactenum}
\end{theorem}

\ktk

\begin{theorem}
$\alpha\geq1$ ise $\xi\mapsto\alpha\cdot\xi$ i\c slemi normaldir.
\end{theorem}

\ktk
\"Orne\u gin
\Sekle{fig:wx} bak\i n.
\begin{figure}
  \centering
\psset{unit=10.8mm,dotsize=6pt}
\begin{pspicture}(10.24,10.24)
\uput[d](0,0){$1$}
\uput[l](0,0){$1$}
\psline(0,0)(10.24,0)
\psline(0,0)(0,10.24)
\psset{linestyle=dotted}
\psline(! 0 5.12)(! 10.24 5.12)
\psline(! 0 5.12 2.56 add)(! 10.24 5.12 2.56 add)
\psline(! 0 5.12 2.56 add 1.28 add)(! 10.24 5.12 2.56 add 1.28 add)
\psline(! 0 5.12 2.56 add 1.28 add 0.64 add)(! 10.24 5.12 2.56 add 1.28 add 0.64 add)
\psline(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add)(! 10.24 5.12 2.56 add 1.28 add 0.64 add 0.32 add)
\psline(0,10.24)(10.24,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psline(! 5.12 0)(! 5.12 10.24)
\psline(! 5.12 2.56 add 0)(! 5.12 2.56 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0)(! 5.12 2.56 add 1.28 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0.64 add 0)(! 5.12 2.56 add 1.28 add 0.64 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0)(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 10.24)
\psline(10.24,0)(10.24,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Xi-axis
\psdots[dotstyle=|]
(2.56,0)
(! 2.56 1.28 add 0)
(! 2.56 1.28 add 0.64 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(5.12, 0)
\uput[ur](2.56,0){$2$}
\uput[ur](! 5.12 1.28 sub 0){$3$}
\uput[ur](! 5.12 0.64 sub 0){$4$}
\uput[ur](5.12,0){$\upomega$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotsize=4pt]
(! 5.12 0.64 add 0)
(! 5.12 0.64 add 0.32 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
\uput[u]{45}(! 5.12 0.64 add 0){$\upomega+1$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 1.28 add 0)
(! 5.12 1.28 add 0.64 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 0)
\uput[u]{45}(! 5.12 1.28 add 0){$\upomega\cdot2$}
\uput[u]{45}(! 5.12 1.28 add 0.64 add 0){$\upomega\cdot3$}
\uput[u]{45}(! 10.24 2.56 sub 0){$\upomega^2$}
\uput[u]{45}(! 10.24 2.56 sub 0.64 add 0){$\upomega^2+\upomega$}
\uput[u]{45}(! 10.24 1.28 sub 0){$\upomega^3$}
\uput[u]{45}(10.24,0){$\upomega^{\upomega}$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 0.64 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.32 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(10.24, 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Eta-axis
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(0,2.56)
(! 0 2.56 1.28 add)
(! 0 2.56 1.28 add 0.64 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(0,5.12)
\uput[ur](0,2.56){$2$}
\uput[ur](! 0 5.12 1.28 sub){$3$}
\uput[ur](! 0 5.12 0.64 sub){$4$}
\uput[ur](0,5.12){$\upomega$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270,dotsize=4pt]
(! 0 5.12 0.64 add)
(! 0 5.12 0.64 add 0.32 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
\uput[ur](! 0 5.12 0.64 add){$\upomega+1$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 1.28 add)
(! 0 5.12 1.28 add 0.64 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add)
\uput[ur](! 0 5.12 1.28 add){$\upomega\cdot2$}
\uput[ur](! 0 10.24 2.56 sub){$\upomega^2$}
\uput[ur](! 0 10.24 1.28 sub){$\upomega^3$}
\uput[ur](0,10.24){$\upomega^{\upomega}$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 0.64 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.32 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(0,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots: [0, w)
\psdots
(! 0             5.12)
(! 2.56          5.12 1.28 add)
(! 5.12 1.28 sub 5.12 2.56 add 0.64 sub)
(! 5.12 0.64 sub 5.12 2.56 add 0.32 sub)
(! 5.12 0.32 sub 5.12 2.56 add 0.16 sub)
(! 5.12 0.16 sub 5.12 2.56 add 0.08 sub)
(! 5.12 0.08 sub 5.12 2.56 add 0.04 sub)
(! 5.12 0.04 sub 5.12 2.56 add 0.02 sub)
(! 5.12 0.02 sub 5.12 2.56 add 0.01 sub)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots: [w, w.2)
\psdots
(! 5.12                                                       5.12 2.56 add)
(! 5.12 0.64 add                                              5.12 2.56 add 0.32 add)
(! 5.12 0.64 add 0.32 add                                     5.12 2.56 add 0.32 add 0.16 add)
(! 5.12 0.64 add 0.32 add 0.16 add                            5.12 2.56 add 0.32 add 0.16 add 0.08 add)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add                   5.12 2.56 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.2, w.3)
\psdots
(! 5.12 1.28 add                                              5.12 2.56 add 0.64 add)
(! 5.12 1.28 add 0.32 add                                     5.12 2.56 add 0.64 add 0.16 add)
(! 5.12 1.28 add 0.32 add 0.16 add                            5.12 2.56 add 0.64 add 0.16 add 0.08 add)
(! 5.12 1.28 add 0.32 add 0.16 add 0.08 add                   5.12 2.56 add 0.64 add 0.16 add 0.08 add 0.04 add)
(! 5.12 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.3, w.4)
\psdots
(! 5.12 1.28 add 0.64 add                                     5.12 2.56 add 0.64 add 0.32 add)
(! 5.12 1.28 add 0.64 add 0.16 add                            5.12 2.56 add 0.64 add 0.32 add 0.08 add)
(! 5.12 1.28 add 0.64 add 0.16 add 0.08 add                   5.12 2.56 add 0.64 add 0.32 add 0.08 add 0.04 add)
(! 5.12 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add)
(! 5.12 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.4, w.5)
\psdots
(! 5.12 1.28 add 0.64 add 0.32 add                            5.12 2.56 add 0.64 add 0.32 add 0.16 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.08 add                   5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.04 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add          5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.5, w.6)
\psdots
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add                   5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add          5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.6, w.7)
\psdots
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add          5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.7, w^2)
\psdots
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2, w^2 + w)
\psdots
(! 5.12 2.56 add                                              5.12 2.56 add 1.28 add)
(! 5.12 2.56 add 0.32 add                                     5.12 2.56 add 1.28 add 0.16 add)
(! 5.12 2.56 add 0.32 add 0.16 add                            5.12 2.56 add 1.28 add 0.16 add 0.08 add)
(! 5.12 2.56 add 0.32 add 0.16 add 0.08 add                   5.12 2.56 add 1.28 add 0.16 add 0.08 add 0.04 add)
(! 5.12 2.56 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2 + w, w^2 + w.2)
\psdots
(! 5.12 2.56 add 0.64 add                                     5.12 2.56 add 1.28 add 0.32 add)
(! 5.12 2.56 add 0.64 add 0.16 add                            5.12 2.56 add 1.28 add 0.32 add 0.08 add)
(! 5.12 2.56 add 0.64 add 0.16 add 0.08 add                   5.12 2.56 add 1.28 add 0.32 add 0.08 add 0.04 add)
(! 5.12 2.56 add 0.64 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.32 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2 + w.2, w^2 + w.3)
\psdots
(! 5.12 2.56 add 0.64 add 0.32 add                            5.12 2.56 add 1.28 add 0.32 add 0.16 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.08 add                   5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.04 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.04 add 0.02 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2 + w.3, w^2 + w.4)
\psdots
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add                            5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.04 add          5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.02 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2 + w.4, w^2 + w.5)
\psdots
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add          5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2 + w.5, w^3)
\psdots
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^3, w^3 + w)
\psdots
(! 5.12 2.56 add 1.28 add                                     5.12 2.56 add 1.28 add 0.64 add)
(! 5.12 2.56 add 1.28 add 0.16 add                            5.12 2.56 add 1.28 add 0.64 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.16 add 0.08 add                   5.12 2.56 add 1.28 add 0.64 add 0.08 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^3 + w, w^3 + w.2)
\psdots
(! 5.12 2.56 add 1.28 add 0.32 add                            5.12 2.56 add 1.28 add 0.64 add 0.16 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.08 add                   5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^3 + w.2, w^3 + w.3)
\psdots
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add                            5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^3 + w.3, w^3 + w.4)
\psdots
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add          5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^3 + w.4, w^4)
\psdots
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^4, w^4 + w)
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add                            5.12 2.56 add 1.28 add 0.64 add 0.32 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.08 add                   5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add                            5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add                   5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add                            5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add                   5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add                   5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add          5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots(! 10.24 0.02 sub 10.24 0.01 sub)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdot(10.24,10.24)
\end{pspicture}

  \caption{$\eta=\upomega\cdot\xi$ denkleminin grafi\u gi}\label{fig:wx}  
\end{figure}
\c Sekilde
\begin{align*}
	\upomega^2&=\upomega\cdot\upomega,&
\upomega^3&=\upomega^2\cdot\upomega,&
\upomega^4&=\upomega^3\cdot\upomega,
\end{align*}
ve genelde,
\Teoremi{thm:rec} kullanan resmi \"ozyineli tan\i ma g\"ore,
\begin{align*}
\alpha^0&=1,&
	\alpha^1&=\alpha,&
	\alpha^{k+1}&=\alpha^k\cdot\alpha.
\end{align*}
Ayr\i ca
\begin{equation*}
\upomega^{\upomega}=\sup_{x<\upomega}(\upomega^x).
\end{equation*}

\begin{xca}
$\xi\mapsto\xi^2$ g\"ondermesi kesin artan m\i d\i r?
S\"urekli midir?
\end{xca}

\begin{theorem}
Ordinal \c carpma, toplama \"uzerine soldan da\u g\i l\i r.
\end{theorem}

\begin{proof}
Ordinal t\"umevar\i m ile
\begin{equation}\label{eqn:dist}
 \alpha\cdot(\beta+\gamma)=\alpha\cdot\beta+\alpha\cdot\gamma
\end{equation}
kan\i tlayaca\u g\i z.
\begin{asparaenum}
\item
$\alpha\cdot(\beta+0)
=\alpha\cdot\beta
=\alpha\cdot\beta+0
=\alpha\cdot\beta+\alpha\cdot0$.
\item
E\u ger \eqref{eqn:dist} do\u gru ise, o zaman
\begin{align*}
	\alpha\cdot(\beta+\gamma')
	&=\alpha\cdot(\beta+\gamma)'\\
	&=\alpha\cdot(\beta+\gamma)+\alpha\\
	&=(\alpha\cdot\beta+\alpha\cdot\gamma)+\alpha\\
	&=\alpha\cdot\beta+(\alpha\cdot\gamma+\alpha)\\
	&=\alpha\cdot\beta+\alpha\cdot\gamma'.
\end{align*}
\item\c Simdi $\gamma$ limit ve
  \begin{equation*}
\Forall{\xi}(\xi<\gamma\lto\alpha\cdot(\beta+\xi)
=\alpha\cdot\beta+\alpha\cdot\xi)
  \end{equation*}
olsun.
E\u ger $\alpha=0$ ise, iddia apa\c c\i kt\i r,
dolay\i s\i yla $\alpha>0$ varsayaca\u g\i z.  
\begin{align*}
	\alpha\cdot(\beta+\gamma)
	&=\alpha\cdot\sup_{\xi<\gamma}(\beta+\xi)&&\text{[tan\i m]}\\
	&=\sup_{\xi<\gamma}\bigl(\alpha\cdot(\beta+\xi)\bigr)&&\text{[$\eta\mapsto\alpha\cdot\eta$ normaldir]}\\
	&=\sup_{\xi<\gamma}(\alpha\cdot\beta+\alpha\cdot\xi)&&\text{[t\"umevar\i m hipotezi]}\\
	&=\alpha\cdot\beta+\sup_{\xi<\gamma}(\alpha\cdot\xi)&&\text{[$\eta\mapsto\alpha\cdot\beta+\eta$ normaldir]}\\
	&=\alpha\cdot\beta+\alpha\cdot\gamma.&&\text{[tan\i m]}\qedhere
\end{align*}
\end{asparaenum}
\end{proof}

\begin{xca}
A\c sa\u g\i daki kan\i t nerede yanl\i\c st\i r?
\begin{falseproof}\noindent
\begin{compactenum}
\item
$0\cdot(\beta+\gamma)=0=0+0=0\cdot\beta+0\cdot\gamma$.
\item
E\u ger \eqref{eqn:dist} do\u gru ise, o zaman
\begin{align*}
\alpha'\cdot(\beta+\gamma)
&=\alpha\cdot(\beta+\gamma)+(\beta+\gamma)\\
&=(\alpha\cdot\beta+\alpha\cdot\gamma)+(\beta+\gamma)\\
&=(\alpha\cdot\beta+\beta)+(\alpha\cdot\gamma+\gamma)\\
&=\alpha'\cdot\beta+\alpha'\cdot\gamma.
\end{align*}
\item
E\u ger $\alpha$ limit ve
$\Forall{\xi}\bigl(\xi<\alpha\lto\xi\cdot(\beta+\gamma)=\xi\cdot\beta+\xi\cdot\gamma\bigr)$ ise
\begin{align*}
	\alpha\cdot(\beta+\gamma)
	&=\sup_{\xi<\alpha}\bigl(\xi\cdot(\beta+\gamma)\bigr)\\
	&=\sup_{\xi<\alpha}\bigl(\xi\cdot\beta+\xi\cdot\gamma)\\
	&=\sup_{\xi<\alpha}(\xi\cdot\beta)+\sup_{\xi<\alpha}(\xi\cdot\gamma)\\
	&=\alpha\cdot\beta+\alpha\cdot\gamma.
\end{align*}
\end{compactenum}
\end{falseproof}
\end{xca}

\begin{xca}
A\c sa\u g\i daki kan\i t nerede yanl\i\c st\i r?
\begin{falseproof}\noindent
\begin{compactenum}
\item
$(\alpha+\beta)\cdot0=0=0+0=\alpha\cdot0+\beta\cdot0$.
\item
E\u ger $(\alpha+\beta)\cdot\gamma=\alpha\cdot\gamma+\beta\cdot\gamma$ ise, o zaman
\begin{align*}
(\alpha+\beta)\cdot\gamma'
&=(\alpha+\beta)\cdot\gamma+(\alpha+\beta)\\
&=(\alpha\cdot\gamma+\beta\cdot\gamma)+(\alpha+\beta)\\
&=(\alpha\cdot\gamma+\alpha)+(\beta\cdot\gamma+\beta)\\
&=\alpha\cdot\gamma'+\beta\cdot\gamma'.
\end{align*}
\item
E\u ger $\gamma$ limit ve
$\Forall{\xi}\bigl(\xi<\gamma\lto(\alpha+\beta)\cdot\xi=\alpha\cdot\xi+\beta\cdot\xi\bigr)$ ise
\begin{align*}
(\alpha+\beta)\cdot\gamma
&=\sup_{\xi<\gamma}\bigl((\alpha+\beta)\cdot\xi\bigr)\\
&=\sup_{\xi<\gamma}(\alpha\cdot\xi+\beta\cdot\xi)\\
&=\sup_{\xi<\gamma}(\alpha\cdot\xi)+\sup_{\xi<\gamma}(\beta\cdot\xi)\\
&=\alpha\cdot\gamma+\beta\cdot\gamma.
\end{align*}
\end{compactenum}
\end{falseproof}
\end{xca}

\begin{theorem}\label{thm:.assoc}
Ordinal \c carpma birle\c smelidir.
\end{theorem}

\ktk

\c Simdi herhangi $n$ sayma say\i s\i\ i\c cin
\begin{equation*}
\alpha^n=\underbrace{\alpha\cdots\alpha}_n
\end{equation*}
anla\c s\i labilir.

\begin{theorem}
$k<\upomega$ ve $\ell<\upomega$ ise
$\alpha^{k+\ell}=\alpha^k\cdot\alpha^{\ell}$.
\end{theorem}

\ktk[ (T\"umevar\i m kullan\i n.)]

\begin{theorem}
Her $\xi\mapsto\xi\cdot\alpha$ i\c slemi artand\i r.
\end{theorem}

\ktk

\section{Hesaplamalar}

\begin{lemma}
	$0<\ell<\upomega$ ise $1+\upomega^{\ell}=\upomega^{\ell}$.
\end{lemma}

\ktk

\begin{theorem}\label{thm:w^k+w^m=w^m}
$k<m<\upomega$ ise $\upomega^k+\upomega^m=\upomega^m$.
\end{theorem}

\begin{proof}
Bir $\ell$ i\c cin, $k+\ell=m$ ve $0<\ell<\upomega$,
dolay\i s\i yla
\begin{align*}
\upomega^k+\upomega^m
&=\upomega^k+\upomega^{k+\ell}\\
&=\upomega^k+\upomega^k\cdot\upomega^{\ell}\\
&=\upomega^k\cdot(1+\upomega^{\ell})\\
&=\upomega^k\cdot\upomega^{\ell}\\
&=\upomega^{k+\ell}\\
&=\upomega^m.\qedhere
\end{align*}
\end{proof}

\begin{theorem}
  $1\leq k<\upomega$ ise $k\cdot\upomega=\upomega$.
(\Sekle{fig:xw} bak\i n.)
\begin{figure}
  \centering
\psset{unit=10.8mm,dotsize=6pt}
\begin{pspicture}(10.24,10.24)
\uput[d](0,0){$1$}
\uput[l](0,0){$1$}
\psline(0,0)(10.24,0)
\psline(0,0)(0,10.24)
\psset{linestyle=dotted}
\psline(! 0 5.12)(! 10.24 5.12)
\psline(! 0 5.12 2.56 add)(! 10.24 5.12 2.56 add)
\psline(! 0 5.12 2.56 add 1.28 add)(! 10.24 5.12 2.56 add 1.28 add)
\psline(! 0 5.12 2.56 add 1.28 add 0.64 add)(! 10.24 5.12 2.56 add 1.28 add 0.64 add)
\psline(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add)(! 10.24 5.12 2.56 add 1.28 add 0.64 add 0.32 add)
\psline(0,10.24)(10.24,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psline(! 5.12 0)(! 5.12 10.24)
\psline(! 5.12 2.56 add 0)(! 5.12 2.56 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0)(! 5.12 2.56 add 1.28 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0.64 add 0)(! 5.12 2.56 add 1.28 add 0.64 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0)(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 10.24)
\psline(10.24,0)(10.24,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Xi-axis
\psdots[dotstyle=|]
(2.56,0)
(! 2.56 1.28 add 0)
(! 2.56 1.28 add 0.64 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(5.12, 0)
\uput[ur](2.56,0){$2$}
\uput[ur](! 5.12 1.28 sub 0){$3$}
\uput[ur](! 5.12 0.64 sub 0){$4$}
\uput[ur](5.12,0){$\upomega$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotsize=4pt]
(! 5.12 0.64 add 0)
(! 5.12 0.64 add 0.32 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
\uput[u]{45}(! 5.12 0.64 add 0){$\upomega+1$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 1.28 add 0)
(! 5.12 1.28 add 0.64 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 0)
\uput[u]{45}(! 5.12 1.28 add 0){$\upomega\cdot2$}
\uput[u]{45}(! 5.12 1.28 add 0.64 add 0){$\upomega\cdot3$}
\uput[u]{45}(! 10.24 2.56 sub 0){$\upomega^2$}
\uput[u]{45}(! 10.24 2.56 sub 0.64 add 0){$\upomega^2+\upomega$}
\uput[u]{45}(! 10.24 1.28 sub 0){$\upomega^3$}
\uput[u]{45}(10.24,0){$\upomega^{\upomega}$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 0.64 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.32 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(10.24, 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Eta-axis
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(0,2.56)
(! 0 2.56 1.28 add)
(! 0 2.56 1.28 add 0.64 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(0,5.12)
\uput[ur](0,2.56){$2$}
\uput[ur](! 0 5.12 1.28 sub){$3$}
\uput[ur](! 0 5.12 0.64 sub){$4$}
\uput[ur](0,5.12){$\upomega$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270,dotsize=4pt]
(! 0 5.12 0.64 add)
(! 0 5.12 0.64 add 0.32 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
\uput[ur](! 0 5.12 0.64 add){$\upomega+1$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 1.28 add)
(! 0 5.12 1.28 add 0.64 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add)
\uput[ur](! 0 5.12 1.28 add){$\upomega\cdot2$}
\uput[ur](! 0 10.24 2.56 sub){$\upomega^2$}
\uput[ur](! 0 10.24 1.28 sub){$\upomega^3$}
\uput[ur](0,10.24){$\upomega^{\upomega}$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 0.64 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.32 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(0,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots: [1, w)
\psdots
(! 0             5.12)
(! 2.56          5.12)
(! 5.12 1.28 sub 5.12)
(! 5.12 0.64 sub 5.12)
(! 5.12 0.32 sub 5.12)
(! 5.12 0.16 sub 5.12)
(! 5.12 0.08 sub 5.12)
(! 5.12 0.04 sub 5.12)
(! 5.12 0.02 sub 5.12)
\psdots[dotstyle=o]
(5.12,5.12)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots: [w, w.2)
\psdots
(! 5.12                                                       5.12 2.56 add)
(! 5.12 0.64 add                                              5.12 2.56 add)
(! 5.12 0.64 add 0.32 add                                     5.12 2.56 add)
(! 5.12 0.64 add 0.32 add 0.16 add                            5.12 2.56 add)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add                   5.12 2.56 add)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add)
\psdots[dotstyle=o]
(! 5.12 1.28 add                                              5.12 2.56 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.2, w.3)
\psdots
(! 5.12 1.28 add                                              5.12 2.56 add 0.64 add)
(! 5.12 1.28 add 0.32 add                                     5.12 2.56 add 0.64 add)
(! 5.12 1.28 add 0.32 add 0.16 add                            5.12 2.56 add 0.64 add)
(! 5.12 1.28 add 0.32 add 0.16 add 0.08 add                   5.12 2.56 add 0.64 add)
(! 5.12 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 0.64 add)
(! 5.12 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 0.64 add)
\psdots[dotstyle=o]
(! 5.12 1.28 add 0.64 add                                     5.12 2.56 add 0.64 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.3, w.4)
\psdots
(! 5.12 1.28 add 0.64 add                                     5.12 2.56 add 0.64 add 0.32 add)
(! 5.12 1.28 add 0.64 add 0.16 add                            5.12 2.56 add 0.64 add 0.32 add)
(! 5.12 1.28 add 0.64 add 0.16 add 0.08 add                   5.12 2.56 add 0.64 add 0.32 add)
(! 5.12 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 0.64 add 0.32 add)
(! 5.12 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 0.64 add 0.32 add)
\psdots[dotstyle=o]
(! 5.12 1.28 add 0.64 add 0.32 add                            5.12 2.56 add 0.64 add 0.32 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.4, w.5)
\psdots
(! 5.12 1.28 add 0.64 add 0.32 add                            5.12 2.56 add 0.64 add 0.32 add 0.16 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.08 add                   5.12 2.56 add 0.64 add 0.32 add 0.16 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add          5.12 2.56 add 0.64 add 0.32 add 0.16 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 0.64 add 0.32 add 0.16 add)
\psdots[dotstyle=o]
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add                   5.12 2.56 add 0.64 add 0.32 add 0.16 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.5, w.6)
\psdots
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add                   5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add          5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add)
\psdots[dotstyle=o]
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add          5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.6, w.7)
\psdots
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add          5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
\psdots[dotstyle=o]
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w.7, w^2)
\psdots
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2, w^2 + w)
\psdots
(! 5.12 2.56 add                                              5.12 2.56 add 1.28 add)
(! 5.12 2.56 add 0.32 add                                     5.12 2.56 add 1.28 add)
(! 5.12 2.56 add 0.32 add 0.16 add                            5.12 2.56 add 1.28 add)
(! 5.12 2.56 add 0.32 add 0.16 add 0.08 add                   5.12 2.56 add 1.28 add)
(! 5.12 2.56 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add)
(! 5.12 2.56 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add)
\psdots[dotstyle=o]
(! 5.12 2.56 add 0.64 add                                     5.12 2.56 add 1.28 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2 + w, w^2 + w.2)
\psdots
(! 5.12 2.56 add 0.64 add                                     5.12 2.56 add 1.28 add 0.32 add)
(! 5.12 2.56 add 0.64 add 0.16 add                            5.12 2.56 add 1.28 add 0.32 add)
(! 5.12 2.56 add 0.64 add 0.16 add 0.08 add                   5.12 2.56 add 1.28 add 0.32 add)
(! 5.12 2.56 add 0.64 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.32 add)
(! 5.12 2.56 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.32 add)
\psdots[dotstyle=o]
(! 5.12 2.56 add 0.64 add 0.32 add                            5.12 2.56 add 1.28 add 0.32 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2 + w.2, w^2 + w.3)
\psdots
(! 5.12 2.56 add 0.64 add 0.32 add                            5.12 2.56 add 1.28 add 0.32 add 0.16 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.08 add                   5.12 2.56 add 1.28 add 0.32 add 0.16 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.32 add 0.16 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.32 add 0.16 add)
\psdots[dotstyle=o]
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add                   5.12 2.56 add 1.28 add 0.32 add 0.16 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2 + w.3, w^2 + w.4)
\psdots
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add                   5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.04 add          5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add)
\psdots[dotstyle=o]
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add          5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2 + w.4, w^2 + w.5)
\psdots
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add          5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add)
\psdots[dotstyle=o]
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^2 + w.5, w^3)
\psdots
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^3, w^3 + w)
\psdots
(! 5.12 2.56 add 1.28 add                                     5.12 2.56 add 1.28 add 0.64 add)
(! 5.12 2.56 add 1.28 add 0.16 add                            5.12 2.56 add 1.28 add 0.64 add)
(! 5.12 2.56 add 1.28 add 0.16 add 0.08 add                   5.12 2.56 add 1.28 add 0.64 add)
(! 5.12 2.56 add 1.28 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add)
(! 5.12 2.56 add 1.28 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add)
\psdots[dotstyle=o]
(! 5.12 2.56 add 1.28 add 0.32 add                            5.12 2.56 add 1.28 add 0.64 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^3 + w, w^3 + w.2)
\psdots
(! 5.12 2.56 add 1.28 add 0.32 add                            5.12 2.56 add 1.28 add 0.64 add 0.16 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.08 add                   5.12 2.56 add 1.28 add 0.64 add 0.16 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.16 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.16 add)
\psdots[dotstyle=o]
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add                   5.12 2.56 add 1.28 add 0.64 add 0.16 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^3 + w.2, w^3 + w.3)
\psdots
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add                   5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add)
\psdots[dotstyle=o]
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add          5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^3 + w.3, w^3 + w.4)
\psdots
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add          5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add)
\psdots[dotstyle=o]
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^3 + w.4, w^4)
\psdots
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots [w^4, w^4 + w)
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add                            5.12 2.56 add 1.28 add 0.64 add 0.32 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.08 add                   5.12 2.56 add 1.28 add 0.64 add 0.32 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.32 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add)
\psdots[dotstyle=o]
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add                   5.12 2.56 add 1.28 add 0.64 add 0.32 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add                            5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add                   5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add                            5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add                   5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add                   5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add          5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add          5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots(! 10.24 0.02 sub 10.24 0.01 sub)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdot(10.24,10.24)
\end{pspicture}

  \caption{$\eta=\xi\cdot\upomega$ denkleminin grafi\u gi}\label{fig:xw}  
\end{figure}

\end{theorem}

\ktk

\begin{theorem}[B\"olme]\label{thm:division}
$1\leq\alpha$ ise $(\xi,\eta)$ i\c cin
\begin{equation}\label{eqn:ax+y=b}
\alpha\cdot\eta+\xi=\beta
\land
\xi<\alpha
\end{equation}
sisteminin bir ve tek bir \c c\"oz\"um\"u vard\i r.
\end{theorem}

\begin{xca}
  Teoremi kan\i tlay\i n.
\"Orne\u gin, a\c sa\u g\i daki iddialar\i\ g\"osterin.
\begin{asparaenum}
  \item
$\alpha>0$ ise $\alpha\cdot\beta\geq\beta$.
\item
$\{\eta\colon\alpha\cdot\eta\leq\beta\}$ k\"umesinin \"usts\i n\i r\i\ vard\i r.
\item
$\sup\{\eta\colon\alpha\cdot\eta\leq\beta\}=\delta$ olsun.
O zaman $\alpha\cdot\gamma+\xi=\beta$ denkleminin 
$\gamma$ \c c\"oz\"um\"u vard\i r, ve $\delta<\alpha$.
Ayr\i ca $(\gamma,\delta)$,
\eqref{eqn:ax+y=b} sisteminin tek \c c\"oz\"um\"u vard\i r.
\end{asparaenum}
\end{xca}

\begin{theorem}
$\upomega^{\upomega}$,
\begin{equation*}
\upomega\cdot\xi=\xi
\end{equation*}
denkleminin en k\"u\c c\"uk \c c\"oz\"um\"ud\"ur.
\end{theorem}

\begin{proof}
$\begin{aligned}[t]
\upomega\cdot\upomega^{\upomega}
&=\upomega\cdot\sup_{x<\upomega}(\upomega^x)\\
&=\sup_{x<\upomega}(\upomega\cdot\upomega^x)\\
&=\sup_{x<\upomega}(\upomega^{1+x})\\
&=\upomega^{\upomega}.\end{aligned}$

\c Simdi $\alpha<\upomega^{\upomega}$ olsun.
O zaman bir $k$ do\u gal say\i s\i\ i\c cin
\begin{gather*}
	\upomega^k\leq\alpha<\upomega^{k+1},\\
	\upomega^{k+1}\leq\upomega\cdot\alpha,
\end{gather*}
dolay\i s\i yla $\alpha<\upomega\cdot\alpha$.
\end{proof}

Teorem sayesinde $\alpha<\upomega^{\upomega}$ ise,
o zaman baz\i\ $\alpha_1$ ve $a_0$ i\c cin
\begin{align*}
	\upomega\cdot\alpha_1+a_0&=\alpha,&
	\alpha_1&<\alpha,&
	a_0&<\upomega.
\end{align*}
E\u ger $\alpha_1>0$ ise,
o zaman baz\i\ $\alpha_2$ ve $a_1$ i\c cin
\begin{align*}
	\upomega\cdot\alpha_2+a_1&=\alpha_1,&
	\alpha_2&<\alpha_1,&
	a_1&<\upomega,
\end{align*}
ve saire.
O zaman bir $k$ i\c cin
\begin{align*}
	\alpha_{k+1}&=0,\\
	\alpha_k&=a_k,\\
	\alpha_{k-1}&=\upomega\cdot a_k+a_{k-1},\\
	\alpha_{k-2}&=\upomega^2\cdot a_k+\upomega\cdot a_{k-1}+a_{k-2},\\
	&\dots\\
	\alpha_1&=\upomega^{k-1}\cdot a_k+\upomega^{k-2}\cdot a_{k-1}+\dots+\upomega\cdot a_2+a_1,\\
	\alpha&=\upomega^k\cdot a_k+\upomega^{k-1}\cdot a_{k-1}+\dots+\upomega^2\cdot a_2+\upomega\cdot a_1+a_0.
\end{align*}
Burada baz\i\ $a_i$ s\i f\i r olabilir.
S\i f\i r terimler silinirse,
o zaman bir $n$ i\c cin, 
\begin{equation*}
\upomega>b_0>b_1>\dots>b_{n-1}
\end{equation*}
ko\c sulunu sa\u glayan
baz\i\ $b_i$ i\c cin, ve baz\i\ $c_i$ sayma say\i lar\i\ i\c cin
\begin{equation*}
\alpha=\upomega^{b_0}\cdot c_0+\upomega^{b_1}\cdot c_1+\dots+\upomega^{b_{n-1}}\cdot c_{n-1}.
\end{equation*}
Bu ifadeye $\alpha$'n\i n \textbf{Cantor normal bi\c cimi} denir.
($0$'\i n Cantor normal bi\c cimi $0$'d\i r.)

\begin{theorem}\label{thm:a+w^m=w^m}
$0<m<\upomega$ ve $\alpha<\upomega^m$ ise $\alpha+\upomega^m=\upomega^m$.
\end{theorem}

\begin{proof}
$\alpha$'n\i n Cantor normal bi\c cimini yaz\i n
ve \Teoremi{thm:w^k+w^m=w^m} kullan\i n.
\end{proof}

\begin{corollary}
$m<\upomega$, $n\in\N$ ve $k\in\N$ ise
\begin{equation*}
(\upomega^m\cdot n+\alpha)\cdot k=\upomega^m\cdot n\cdot k+\alpha.
\end{equation*}
\end{corollary}

\ksk

\"Orne\u gin
\begin{equation*}
(\upomega^5\cdot10+\upomega^3\cdot8+\upomega)\cdot6
=\upomega^5\cdot60+\upomega^3\cdot8+\upomega.
\end{equation*}

Sonucun durumunda a\c sa\u g\i daki e\c sitlik \c c\i kar.
\begin{align*}
(\upomega^m\cdot n+\alpha)\cdot\upomega
&=\upomega^m\cdot n+\underbrace{\alpha+\upomega^m\cdot n}_{\upomega^m\cdot n}+\underbrace{\alpha+\upomega^m\cdot n}_{\upomega^m\cdot n}+\cdots\\
&=\upomega^m\cdot n\cdot\upomega\\
&=\upomega^{m+1}.
\end{align*}
Asl\i nda e\c sitli\u gin ger\c cek kan\i t\i n\i n
\Teoreme{thm:a+w^m=w^m} ihtiyac\i\ yoktur.

\begin{theorem}
$m<\upomega$, $n\in\N$ ve $\alpha<\upomega^m$ ise
\begin{equation*}
(\upomega^m\cdot n+\alpha)\cdot\upomega=\upomega^{m+1},
\end{equation*}
dolay\i s\i yla
$k\in\N$ ise
\begin{equation*}
(\upomega^m\cdot n+\alpha)\cdot\upomega^k=\upomega^{m+k}.
\end{equation*}
\end{theorem}

\begin{proof}
$(\upomega^m\cdot n+\alpha)\cdot k
<\upomega^m\cdot(n+1)\cdot k$
oldu\u gundan
\begin{align*}
	\upomega^{m+1}
	&\leq(\upomega^m\cdot n+\alpha)\cdot\upomega\\
  &=\sup_{x<\upomega}\bigl((\upomega^m\cdot n+\alpha)\cdot x\bigr)\\
  &\leq\sup_{x<\upomega}\bigl(\upomega^m\cdot(n+1)\cdot x\bigr)\\
	&=\upomega^{m+1}.\qedhere
\end{align*}
\end{proof}

\"Orne\u gin
\begin{align*}
	&\phantom{{}={}}(\upomega^3\cdot4+\upomega\cdot6)\cdot(\upomega^2\cdot3+8)\\
	&=(\upomega^3\cdot4+\upomega\cdot6)\cdot\upomega^2\cdot3+(\upomega^3\cdot4+\upomega\cdot6)\cdot8\\
	&=\upomega^5\cdot3+\upomega^3\cdot32+\upomega\cdot6.
\end{align*}

\begin{xca}
$(\upomega^9\cdot9+\upomega^2\cdot9+\upomega\cdot9+9)\cdot(\upomega^2\cdot9+\upomega\cdot9+9)$
\c carp\i m\i n\i n Cantor normal bi\c cimini hesaplay\i n.
\end{xca}

\section{Kardinaller}

\begin{theorem}\label{thm:ab=axb}
  $\alpha\cdot\beta
\approx\alpha\times\beta$.
\end{theorem}

\begin{proof}
  \Teoremde{thm:division}n
	\begin{equation*}
	(\xi,\eta)\mapsto\alpha\cdot\eta+\xi,
	\end{equation*}
$\alpha\times\beta$
kartezyan
\c carp\i m\i ndan
$\alpha\cdot\beta$ ordinal \c carp\i m\i na giden
bir e\c slemedir.
\end{proof}

\begin{theorem}
$\card{\upomega\cdot\upomega}=\upomega$.
\end{theorem}

\ktk

\begin{theorem}
$\{\xi\colon\upomega\leq\xi<\upomega^{\upomega}\}$ k\"umesinin her eleman\i n\i n kardinali
$\upomega$'d\i r.
\end{theorem}

\ktk

\begin{theorem}\label{thm:cntbl-u}
Her $k$ do\u gal say\i s\i\ i\c cin $f_k$
bir $A_k$ k\"umesini $\upomega$'ya g\"oms\"un.
O zaman
\begin{equation*}
\bigcup_{i<\upomega}A_i\preccurlyeq\upomega.
\end{equation*}
\end{theorem}

\begin{proof}
$\bigcup_{i<\upomega}A_i$ bile\c siminde
\begin{equation*}
g(x)=\min\{i\colon x\in A_i\}
\end{equation*}
olsun.  O zaman $x\mapsto\left(g(x),f_{g(x)}(x)\right)$ g\"ondermesi,
bile\c simin $\upomega\times\upomega$ \c carp\i m\i na bir g\"ommesidir.
\end{proof}

\begin{corollary}
$\upomega^{\upomega}\approx\upomega$.
\end{corollary}

\begin{proof}
Her $n$ i\c cin, $\upomega^{n+1}=\upomega^n\cdot\upomega$ oldu\u gundan,
\Teoremin{thm:ab=axb} kan\i t\i n\i ndan
kesin bir $f_n$ i\c cin
\begin{equation*}
f_n\colon\upomega^{n+1}\xrightarrow{\approx}\upomega^n\times\upomega.
\end{equation*}
\c Simdi $g\colon\upomega\times\upomega\xrightarrow{\approx}\upomega$ olsun.
O zaman
\begin{equation*}
  g\circ f_1\colon\upomega^2\xrightarrow{\approx}\upomega.
\end{equation*}
M\"umk\"unse
\begin{equation}\label{eqn:hm}
h_m\colon\upomega^m\xrightarrow{\approx}\upomega
\end{equation}
olsun.
O zaman bir ve tek bir $h_{m+1}$ i\c cin,
\begin{gather*}
h_{m+1}\colon\upomega^{m+1}\to\upomega,\\  
\Forall{\xi}\Forall{\eta}\Forall z
\Bigl(f_n(\xi)=(\eta,z)\lto h_{m+1}(\xi)=g\bigl(h_m(\eta),z\bigr)\Bigr);
\end{gather*}
ve bu durumda
\begin{equation*}
  h_{m+1}\colon\upomega^{m+1}\xrightarrow{\approx}\upomega.
\end{equation*}
T\"umevar\i m ve \"ozyinelemeyle her $m$ sayma say\i s\i\ i\c cin,
bir ve tek bir $h_m$ i\c cin, \eqref{eqn:hm} do\u grudur.
\c Simdi
\begin{equation*}
\upomega^{\upomega}=\sup_{0<x<\upomega}(\upomega^x)=\bigcup_{0<x<\upomega}\upomega^x
\end{equation*}
oldu\u gundan teoremi kullan\i labilir.
\end{proof}

\chapter{Ordinal kuvvet alma}\label{ch:exp}

\section{Tan\i m ve \"ozellikler}

\"Ozyineli tan\i ma g\"ore her $\alpha$ i\c cin
\begin{gather*}
	\alpha^0=1,\\
	\alpha^{\beta'}=\alpha^\beta\cdot\alpha,\\
	\gamma\text{ limit ise }\alpha^\gamma
=\sup_{0<\xi<\gamma}(\alpha^{\xi})
=\limsup_{\xi\to\gamma^-}(\alpha^{\xi}).
\end{gather*}

\begin{theorem}
  $\alpha^1=\alpha$, $1^{\alpha}=1$, ve
  \begin{equation*}
  0^{\alpha}=
  \begin{cases}
    1,&\text{ $\alpha=0$ durumunda},\\
0,&\text{ $\alpha>0$ durumunda}.
  \end{cases}
  \end{equation*}
\end{theorem}

\ktk

\begin{theorem}
$\alpha\geq2$ ise $\xi\mapsto\alpha^{\xi}$ i\c slemi, normaldir.
\end{theorem}

\ktk

\Sekle{fig:w^x} bak\i n.
\begin{figure}
  \centering
\psset{unit=10.8mm,dotsize=6pt}
\begin{pspicture}(10.24,10.24)
\uput[d](0,0){$1$}
\uput[l](0,0){$1$}
\psline(0,0)(10.24,0)
\psline(0,0)(0,10.24)
\psline[linewidth=6pt](! 6.68 3.34 5.12 add)(10.24,10.24)
\psset{linestyle=dotted}
\psline(! 0 5.12)(! 10.24 5.12)
\psline(! 0 5.12 2.56 add)(! 10.24 5.12 2.56 add)
\psline(! 0 5.12 2.56 add 1.28 add)(! 10.24 5.12 2.56 add 1.28 add)
\psline(! 0 5.12 2.56 add 1.28 add 0.64 add)(! 10.24 5.12 2.56 add 1.28 add 0.64 add)
\psline(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add)(! 10.24 5.12 2.56 add 1.28 add 0.64 add 0.32 add)
\psline(0,10.24)(10.24,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psline(! 5.12 0)(! 5.12 10.24)
\psline(! 5.12 2.56 add 0)(! 5.12 2.56 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0)(! 5.12 2.56 add 1.28 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0.64 add 0)(! 5.12 2.56 add 1.28 add 0.64 add 10.24)
\psline(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0)(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 10.24)
\psline(10.24,0)(10.24,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Xi-axis
\psdots[dotstyle=|]
(2.56,0)
(! 2.56 1.28 add 0)
(! 2.56 1.28 add 0.64 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(5.12, 0)
\uput[ur](2.56,0){$2$}
\uput[ur](! 5.12 1.28 sub 0){$3$}
\uput[ur](! 5.12 0.64 sub 0){$4$}
\uput[ur](5.12,0){$\upomega$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotsize=4pt]
(! 5.12 0.64 add 0)
(! 5.12 0.64 add 0.32 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
\uput[u]{45}(! 5.12 0.64 add 0){$\upomega\cdot2$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 1.28 add 0)
(! 5.12 1.28 add 0.64 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 0)
\uput[u]{45}(! 5.12 1.28 add 0){$\upomega^2$}
\uput[u]{45}(! 5.12 1.28 add 0.64 add 0){$\upomega^3$}
\uput[u]{45}(! 10.24 2.56 sub 0){$\upomega^{\upomega}$}
\uput[u]{45}(! 10.24 2.56 sub 0.64 add 0){$\upomega^{{\upomega}^2}$}
\uput[u]{45}(! 10.24 1.28 sub 0){$\upomega^{\upomega^{\upomega}}$}
\uput[u]{45}(10.24,0){$\upvarepsilon_0$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 0.64 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.32 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add 0)
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(! 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|]
(10.24, 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Eta-axis
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(0,2.56)
(! 0 2.56 1.28 add)
(! 0 2.56 1.28 add 0.64 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 2.56 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(0,5.12)
\uput[ur](0,2.56){$2$}
\uput[ur](! 0 5.12 1.28 sub){$3$}
\uput[ur](! 0 5.12 0.64 sub){$4$}
\uput[ur](0,5.12){$\upomega$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270,dotsize=4pt]
(! 0 5.12 0.64 add)
(! 0 5.12 0.64 add 0.32 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
\uput[ur](! 0 5.12 0.64 add){$\upomega\cdot2$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 1.28 add)
(! 0 5.12 1.28 add 0.64 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add)
\uput[ur](! 0 5.12 1.28 add){$\upomega^2$}
\uput[ur](! 0 10.24 2.56 sub){$\upomega^{\upomega}$}
\uput[ur](! 0 10.24 1.28 sub){$\upomega^{\upomega^{\upomega}}$}
\uput[ur](0,10.24){$\upvarepsilon_0$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 0.64 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.32 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.08 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.04 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.02 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.01 add)
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(! 0 5.12 2.56 add 1.28 add 0.64 add 0.32 add 0.16 add 0.08 add 0.04 add 0.02 add 0.01 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\psdots[dotstyle=|,dotangle=270]
(0,10.24)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots: 1, 2, 3, 4, ..., w
\psdots(0,5.12)
(! 2.56          1.28          5.12 add)
(! 5.12 1.28 sub 2.56 0.64 sub 5.12 add)
(! 5.12 0.64 sub 2.56 0.32 sub 5.12 add)
(! 5.12 0.32 sub 2.56 0.16 sub 5.12 add)
(! 5.12 0.16 sub 2.56 0.08 sub 5.12 add)
(! 5.12 0.08 sub 2.56 0.04 sub 5.12 add)
(! 5.12 0.04 sub 2.56 0.02 sub 5.12 add)
(! 5.12 0.02 sub 2.56 0.01 sub 5.12 add)
(! 5.12          2.56          5.12 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots w^2, w^3, ...
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots w.2, w.3, ...
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots w, w+1, w+2, ..., w.2
\psdots%(! 5.12 2.56 5.12 add)
(! 5.44 2.72 5.12 add)
(! 5.60 2.80 5.12 add)
(! 5.68 2.84 5.12 add)
(! 5.72 2.86 5.12 add)
(! 5.74 2.87 5.12 add)
(! 5.76 2.88 5.12 add)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dots w.2, w.2+1, w.2+2, ..., w.3
\psdots%(! 5.76 2.88 5.12 add)
(! 5.92 2.96 5.12 add)
(! 6.00 3.00 5.12 add)
(! 6.04 3.02 5.12 add)
(! 6.06 3.03 5.12 add)
(! 6.08 3.04 5.12 add)
% Dots w.3+1, w.3+2, ..., w.4
\psdots%(! 6.08 3.04 5.12 add)
(! 6.16 3.08 5.12 add)
(! 6.20 3.12 5.12 add)
(! 6.22 3.13 5.12 add)
(! 6.24 3.12 5.12 add)
% Dots w.4, w.4+1, ...,w.5
\psdots%(! 6.24 3.12 5.12 add)
(! 6.28 3.14 5.12 add)
(! 6.30 3.15 5.12 add)
(! 6.32 3.16 5.12 add)
% Dots w.5, w.5+1, ..., w.6
\psdots%(! 6.32 3.16 5.12 add)
(! 6.34 3.17 5.12 add)
(! 6.36 3.18 5.12 add)
% Dots w.6, w.6+1, ..., w.7
\psdots%(! 6.36 3.18 5.12 add)
(! 6.38 3.19 5.12 add)
% Dot w.w = w^2
\psdot(! 6.40 3.20 5.12 add)
%%%%%%%%%%%%%
% Dots w^2, w^2+1, ..., w^2+w
\psdots%(! 6.40 3.20 5.12 add)
(! 6.48 3.24 5.12 add)
(! 6.52 3.26 5.12 add)
(! 6.54 3.27 5.12 add)
(! 6.56 3.28 5.12 add)
% Dots w^2+w, w^2+w+1, ..., w^2+w.2
\psdots%(! 6.56 3.28 5.12 add)
(! 6.60 3.30 5.12 add)
(! 6.62 3.31 5.12 add)
(! 6.64 3.32 5.12 add)
% Dots w^2+w.2, w^2+w.2+1, ..., w^2+w.3
\psdots%(! 6.64 3.32 5.12 add)
(! 6.66 3.33 5.12 add)
(! 6.68 3.34 5.12 add)
\psdot(! 10.24 5.12 5.12 add)
\end{pspicture}

  \caption{$\eta=\upomega^{\xi}$ denkleminin grafi\u gi}\label{fig:w^x}  
\end{figure}
\c Sekilde
\begin{equation*}\label{epsilon}
\upvarepsilon_0=\sup\left\{\upomega,\upomega^{\upomega},\upomega^{\upomega^{\upomega}},\dots\right\}.
\end{equation*}

\begin{xca}
$\xi\mapsto\xi^{\xi}$ i\c slemi kesin artan m\i d\i r?
S\"urekli midir?
\end{xca}

\begin{theorem}
$\alpha^{\beta+\gamma}=\alpha^{\beta}\cdot\alpha^{\gamma}$
ve
$\alpha^{\beta\cdot\gamma}=(\alpha^{\beta})^{\gamma}$.
\end{theorem}

\ktk

\begin{theorem}
$\alpha\geq1$ ise $\xi\mapsto\xi^{\alpha}$ artand\i r.
\end{theorem}

\ktk

\section{Hesaplamalar}

\begin{theorem}[Logaritma alma]
$2\leq\alpha$ ve $1\leq\beta$ ise $(\xi,\eta,\zeta)$ i\c cin
\begin{equation*}
\alpha^{\xi}\cdot\eta+\zeta=\beta\land0<\eta<\alpha\land\zeta<\alpha^{\xi}
\end{equation*}
sisteminin bir ve tek bir \c c\"oz\"um\"u vard\i r.
\end{theorem}

\ktk

Teorem sayesinde $1\leq\alpha$ ise,
baz\i\ $\alpha_0$, $a_0$, ve $\beta_1$ i\c cin
\begin{align*}
	\upomega^{\alpha_0}\cdot a_0+\beta_1&=\alpha,&
	0<a_0&<\upomega,&
	\beta_1&<\upomega^{\alpha_0}.
\end{align*}
E\u ger $1\leq\beta_1$ ise, o zaman
baz\i\ $\alpha_1$, $a_1$, ve $\beta_2$ i\c cin
\begin{align*}
	\upomega^{\alpha_1}\cdot a_1+\beta_2&=\beta_1,&
	0<a_1&<\upomega,&
	\beta_2&<\upomega^{\alpha_1},
\end{align*}
ve saire.  O zaman bir $k$ i\c cin
\begin{gather*}
\alpha_0>\alpha_1>\dots>\alpha_k,\\
\{a_0,\dots,a_k\}\included\N,\\
\alpha=\upomega^{\alpha_0}\cdot a_0+\upomega^{\alpha_1}\cdot a_1+\dots+\upomega^{\alpha_k}\cdot a_k.
\end{gather*}
Son ifade, $\alpha$'n\i n \textbf{Cantor normal bi\c cimidir.}

\begin{theorem}
$\alpha<\upomega^{\beta}$ ise $\alpha+\upomega^{\beta}=\upomega^{\beta}$.
\end{theorem}

\ktk[ (\Teoreme{thm:a+w^m=w^m} bak\i n.)]

\begin{corollary}
$\alpha<\upomega^{\beta}$, $n\in\N$, ve $k\in\N$ ise
\begin{equation*}
(\upomega^{\beta}\cdot n+\alpha)\cdot k=\upomega^{\beta}\cdot n\cdot k.
\end{equation*}
\end{corollary}

\ksk

\begin{theorem}
$\alpha<\upomega^{\beta}$, $n\in\N$, ve $1\leq\gamma$ ise
\begin{equation*}
(\upomega^{\beta}\cdot n+\alpha)\cdot\upomega^{\gamma}=\upomega^{\beta+\gamma}.
\end{equation*}
\end{theorem}

\ktk[ (Bir $\delta$ i\c cin $\gamma=1+\delta$ oldu\u gunu kullanabiliriz.)]

\c Simdi iki Cantor normal bi\c ciminin \c carp\i m\i n\i n 
Cantor normal bi\c cimini hesaplayabiliriz.

\begin{theorem}
  $0<k<\upomega$ ise
  \begin{equation*}
    k^{\upomega^{\xi}}=
    \begin{cases}
      k,&\text{ $\xi=0$ durumunda,}\\
\upomega^{\upomega^{\xi-1}},&\text{ $0<\xi<\upomega$ durumunda,}\\
\upomega^{\upomega^{\xi}},&\text{ $\upomega\leq\xi$ durumunda.}
    \end{cases}
  \end{equation*}
\end{theorem}

\ktk

\begin{theorem}
  $\alpha<\upomega^{\beta}$, $n\in\N$, ve $\gamma$ limit ise
  \begin{equation*}
    (\upomega^{\beta}\cdot n+\alpha)^{\gamma}=\upomega^{\beta\cdot\gamma}.
  \end{equation*}
\end{theorem}

\begin{theorem}
$\upvarepsilon_0$,
\begin{equation*}
\upomega^{\xi}=\xi
\end{equation*}
denkleminin en k\"u\c c\"uk \c c\"oz\"um\"ud\"ur.
\end{theorem}

\ktk

\section{Kardinaller}

Herhangi $\alpha$ ve $\beta$ ordinalleri i\c cin,
$\beta$'dan $\alpha$'ya giden g\"ondermeler,
\begin{equation*}
{}^{\beta}\alpha
\end{equation*}
s\i n\i f\i n\i\ olu\c stursun, ve
\begin{equation*}
\exp(\alpha,\beta)=\bigl\{f\colon f\in{}^{\beta}\alpha\land\{\xi\in\beta\colon f(\xi)>0\}\prec\upomega\bigr\}
\end{equation*}
olsun.

\begin{theorem}
$\alpha^{\beta}\approx\exp(\alpha,\beta)$.
\end{theorem}

\begin{proof}
$\exp(1,\beta)\approx1=1^{\beta}$; ayr\i ca
\begin{equation*}
\exp(0,\beta)\approx
\begin{cases}
	1,&\text{ $\beta=0$ durumunda,}\\
	0,&\text{ $\beta>0$ durumunda,}
\end{cases}
\end{equation*}
dolay\i s\i yla $\exp(0,\beta)\approx 0^{\beta}$.
\c Simdi $\alpha\geq2$ olsun.
E\u ger $\gamma<\alpha^\beta$ ise,
o zaman Cantor normal bi\c cimi gibi,
baz\i\ $n$ do\u gal say\i s\i\ i\c cin,
baz\i\ $\gamma_i$ ve $\delta_i$ i\c cin,
\begin{gather*}
	\beta>\gamma_0>\dots>\gamma_{n-1},\\
	\{\delta_i\colon i<n\}\included\alpha\setminus\{0\},\\
	\gamma=\alpha^{\gamma_0}\cdot c_0+\dots+\alpha^{\gamma_{n-1}}\cdot c_{n-1}.
\end{gather*}
\c Simdi tan\i ma g\"ore
\begin{equation*}
f_{\gamma}(\xi)=
\begin{cases}
	\delta_i,&\text{ $\xi=\gamma_i$ durumunda},\\
0,&\text{ $\xi\in\beta\setminus\{\gamma_i\colon i<n\}$ durumunda}
\end{cases}
\end{equation*}
olsun.
O zaman $f_{\gamma}\in\exp(\alpha,\beta)$.
As\i nda
\begin{equation*}
\xi\mapsto f_{\xi}\colon\alpha^{\beta}\xrightarrow{\approx}\exp(\alpha,\beta).\qedhere
\end{equation*}
\end{proof}

\begin{theorem}
$\upvarepsilon_0\approx\upomega$.
\end{theorem}

\ktk

\chapter{Kardinal kuvvetler}\label{ch:card}

\section{Say\i lamaz k\"umeler}

E\u ger $A\preccurlyeq\upomega$ ise,
o zaman $A$ \textbf{say\i labilir;}
di\u ger durumda $A$ \textbf{say\i lamaz.}
G\"ord\"u\u g\"um\"uz gibi
say\i labilir k\"umelerden
ordinal toplama, \c carpma, ve kuvvet alma ile
say\i lamaz k\"umeler elde edilemez.


Herhangi $\bm A$ s\i n\i f\i\ i\c cin
\begin{equation*}\label{pow}
\pow{\bm A},
\end{equation*}
$\bm A$'n\i n \emph{altk\"umeleri} taraf\i ndan olu\c sturulmu\c s s\i n\i ft\i r.
Yani
\begin{equation*}
\pow{\bm A}=\{X\colon X\included\bm A\}.
\end{equation*}
Buradaki $X$ siyah olmad\i\u g\i ndan \emph{k\"ume} de\u gi\c skenidir.
K\"ume olmayan bir s\i n\i f, bir s\i n\i f\i n eleman\i\ olamaz.
E\u ger $\universe$, t\"um k\"umeler taraf\i ndan olu\c sturulmu\c s s\i n\i f ise,
o zaman
\begin{equation*}
\pow{\universe}=\universe.
\end{equation*}
Ama $n\in\upomega$ ise
\begin{equation*}
n<2^n=\card{\pow n}.
\end{equation*}

\begin{theorem}[Cantor]\label{thm:pow}
Her $A$ k\"umesi i\c cin
\begin{equation*}
A\prec\pow A.
\end{equation*}
\end{theorem}

\begin{proof}
$x\mapsto\{x\}\colon A\xrightarrow{\preccurlyeq}\pow A$.
\c Simdi $f\colon A\xrightarrow{\preccurlyeq}\pow A$ ise
\begin{equation*}
B=\{x\in A\colon x\notin f(x)\}
\end{equation*}
olsun.  O zaman $A$'n\i n her $c$ eleman\i\ i\c cin
\begin{equation*}
c\in B\liff c\notin f(c),
\end{equation*}
dolay\i s\i yla $B\neq f(c)$.
Bu \c sekilde $f$, e\c sleme olamaz.
\end{proof}

\begin{xca}
Cantor Teoreminin kan\i t\i\
$A$'n\i n k\"ume oldu\u gunu nas\i l kullan\i r?
\end{xca}

\begin{axiom}[Kuvvet K\"ume]
Her $A$ k\"umesi i\c cin $\pow A$ s\i n\i f\i\ bir k\"umedir.
\end{axiom}

Herhangi $a$ ve $b$ i\c cin
\begin{equation*}
(a,b)=\bigl\{\{a\},\{a,b\}\bigr\}
\end{equation*}
olsun.

\begin{theorem}
$(a,b)=(c,d)$ ancak ve ancak $a=c$ ve $b=d$.
\end{theorem}

\ktk

\c Simdi $A\times B=\{(x,y)\colon x\in A\land y\in B\}$
tan\i mlanabilir.

\begin{theorem}
$A\times B\included\pow{\pow{A\cup B}}$.
\end{theorem}

\ktk

\begin{theorem}[Hartogs]
Her kardinalin daha b\"uy\"u\u g\"u vard\i r.
\end{theorem}

\begin{proof}
$\bm A=\{\xi\colon\xi\preccurlyeq\kappa\}$ olsun.
O zaman $\bm A\included\on$, ve ayr\i ca $\bm A$ ge\c ci\c slidir,
dolay\i s\i yla $\bm A$ bir k\"umeyse, bir $\alpha$ ordinalidir.
Bu durumda $\alpha\notin\alpha$ oldu\u gundan $\alpha>\kappa$.

E\u ger $f$ bir $\beta$'y\i\ $\kappa$'ya g\"om\"urse,
o zaman bir
\begin{equation*}
\Bigl\{\bigl(f(\xi),f(\eta)\bigr)\colon\xi\leq\eta<\beta\Bigr\}
\end{equation*}
k\"umesi
elde edilebilir.
Bu \c sekilde elde edilen t\"um k\"umeler,
$\kappa\times\kappa$ \c carp\i m\i n\i n bir $B$ altk\"umesini olu\c sturur.
O halde $B\approx\bm A$ (neden?),
dolay\i s\i yla $\bm A$ da bir k\"umedir.
\end{proof}

Sonu\c c olarak
\begin{equation*}
\kappa^+=\min\{\xi\colon\kappa<\xi\}
\end{equation*}
tan\i mlanabilir, $\kappa^+$, $\kappa$'n\i n \emph{kardinal} ard\i l\i d\i r.

\c Simdi \"ozyineli tan\i ma g\"ore
\begin{gather*}
	\aleph_0=\upomega,\\
	\aleph_{\alpha'}=(\aleph_{\alpha})^+,\\
	\alpha\text{ limit ise }\aleph_{\alpha}=\sup_{\xi<\alpha}\aleph_{\xi}.
\end{gather*}
(Burada $\aleph$, \.Ibrani \emph{alef} harfidir.)

\begin{theorem}
$\xi\mapsto\aleph_{\xi}$ normaldir,
ve her sonsuz kardinal, bir $\alpha$ i\c cin, $\aleph_{\alpha}$'d\i r.
\end{theorem}

\ktk

\begin{lemma}
  Her sonsuz kardinal, $\upomega$'n\i n bir kuvvetidir.
\end{lemma}

\klk

Tan\i ma g\"ore
\begin{align*}
  \kappa\oplus\lambda&=\card{\kappa\sqcup\lambda}=\card{\kappa+\lambda},\\
  \kappa\otimes\lambda&=\card{\kappa\times\lambda}=\card{\kappa\cdot\lambda}
\end{align*}
olsun;
bunlar $\kappa$ ve $\lambda$'n\i n \textbf{kardinal toplam\i}
ve \textbf{kardinal \c carp\i m\i d\i r.}

\begin{theorem}
E\u ger $\kappa$ ve $\lambda$'n\i n biri sonsuz ise
  \begin{equation*}
    \kappa\oplus\lambda=\max(\kappa,\lambda).
  \end{equation*}
E\u ger
$\kappa$ ve $\lambda$'n\i n biri sonsuz ise ve di\u geri s\i f\i r de\u gilse
  \begin{equation*}
    \kappa\otimes\lambda=\max(\kappa,\lambda).
  \end{equation*}
\end{theorem}

\begin{proof}
$\kappa\leq\lambda$ olsun.  O zaman
  \begin{equation*}
   \lambda
\leq\kappa+\lambda
\leq\lambda+\lambda
\preccurlyeq2\cdot\lambda
\leq\lambda\cdot\lambda,
  \end{equation*}
ve $\kappa>0$ ise
\begin{equation*}
  \lambda\leq\kappa\cdot\lambda\leq\lambda\cdot\lambda,
\end{equation*}
dolay\i s\i yla $\lambda\approx\lambda^2$ kan\i tlamak yeter.

Lemmadan bir $\alpha$ i\c cin $\lambda=\upomega^{\alpha}$.
O zaman $\lambda\approx\exp(\upomega,\alpha)$.
\c Simdi $f\colon\upomega\times\upomega\xrightarrow{\approx}\upomega$ olsun.
E\u ger $g$ ve $h$, $\exp(\upomega,\alpha)$ k\"umesinin eleman\i\ ise $g*h$,
\begin{equation*}
  \xi\mapsto f(g(\xi),h(\xi))
\end{equation*}
eleman\i\ olsun.
O zaman
\begin{equation*}
  (g,h)\mapsto g*h\colon\exp(\upomega,\alpha)\xrightarrow{\approx}
\exp(\upomega,\alpha)\times\exp(\upomega,\alpha).\qedhere
\end{equation*}
\end{proof}

Sonu\c c olarak
\begin{equation*}
  \card{\aleph_{\alpha}+\aleph_{\beta}}=\aleph_{\max(\alpha,\beta)}
=\card{\aleph_{\alpha}\cdot\aleph_{\beta}}.
\end{equation*}

\c Simdi herhangi $A$ k\"umesi i\c cin
\begin{equation*}\label{powf}
  \powf A=\{X\in\pow A\colon\card X<\upomega\}
\end{equation*}
olsun.

\begin{theorem}
E\u ger $\kappa$ sonsuz ise $\powf{\kappa}\approx\kappa$.
\end{theorem}

\begin{proof}
Her $m$ i\c cin
  $\{\xi\in\kappa\colon\card{\xi}=m\}\preccurlyeq\kappa^m\approx\kappa$,
dolay\i s\i yla
\begin{equation*}
  \powf{\kappa}=\bigcup_{i\in\upomega}\{\xi\in\kappa\colon\card{\xi}=i\}
\preccurlyeq\upomega\times\kappa\approx\kappa.\qedhere
\end{equation*}
\end{proof}

\begin{theorem}
E\u ger $\beta$ sonsuz ve $2\leq\alpha\leq\beta$ ise
\begin{equation*}
  \card{\alpha^{\beta}}=\card{\beta}.
\end{equation*}
E\u ger $\alpha$ sonsuz ve $1\leq\beta\leq\alpha$ ise
\begin{equation*}
  \card{\alpha^{\beta}}=\card{\alpha}.
\end{equation*}
  \end{theorem}

\ktk

\section{Se\c cme}

\Teoremde{thm:cntbl-u}, 
$\bigcup_{i<\upomega}A_i$ bile\c siminin say\i labilir olmas\i\ i\c cin,
her $A_k$ k\"umesinin say\i labilir olmas\i\ yetmez,
ama $A_k$ k\"umesinin $\upomega$'ya kesin bir g\"ommesi bilinmelidir.
Her $k$ i\c cin,
$A_k$ k\"umesinin $\upomega$'ya g\"ommeleri,
bo\c s olmayan bir $\mathscr B_k$ k\"umesini olu\c sturabilirler;\label{curly-B}
ama g\"ord\"u\u g\"um\"uz aksiyomlarla
\begin{equation*}
\Forall x(x\in\upomega\lto f_x\in\mathscr B_x)
\end{equation*}
ko\c sulunu sa\u glayan 
$x\mapsto f_x$ g\"ondermesinin olup olmad\i\u g\i n\i\ bilmiyoruz.

G\"ord\"u\u g\"um\"uz aksiyomlar, 
\textbf{Zermelo--Fraenkel} veya \textbf{ZF} aksiyomlar\i d\i r.

Her $k$ i\c cin $\mathscr B_k$ k\"umesinden bir $f_k$ se\c cmek isteriz.
\emph{Se\c cim Aksiyomunun} 
bi\c cimlerinin birine g\"ore, bu se\c cme m\"umk\"und\"ur.
Bizim i\c cin, a\c sa\u g\i daki bi\c cim kullan\i l\i\c sl\i\ olacakt\i r.

\begin{axiom}[Se\c cim]
  Her k\"ume iyis\i ralanabilir.
\end{axiom}

\"Orne\u gin $\bigcup_{x\in\upomega}\mathscr B_x$ iyis\i ralan\i rsa,
o zaman istedi\u gimiz $x\mapsto f_x$ g\"ondermesi
\begin{equation*}
  x\mapsto\min\mathscr(B_x)
\end{equation*}
olabilir.

G\"odel'in kan\i tlad\i\u g\i\ teoreme g\"ore, 
ZF aksiyomlar\i n\i n bir \emph{modelinde,}
Se\c cim Aksiyomu do\u grudur.
Cohen'in kan\i tlad\i\u g\i\ teoreme g\"ore, 
ZF aksiyomlar\i n\i n bir modelinde,
Se\c cim Aksiyomu yanl\i\c st\i r.
K\i saca Se\c cim Aksiyomu, ZF'den ba\u g\i ms\i zd\i r.

\begin{sloppypar}
  Se\c cim Aksiyomunu varsay\i yoruz.
  Bununla ZF, \textbf{ZFC}'dir.
  \c Simdi her k\"umenin kardinali vard\i r.
  Tan\i ma g\"ore
  \begin{equation*}
    \kappa^{\lambda}=\card{{}^{\kappa}\lambda}.
  \end{equation*}
  Bu kuvvet, ordinal k\"uvvet de\u gil,
  \textbf{kardinal kuvvettir.}
  Orne\u gin
  $\aleph_0=\upomega$ oldu\u gu halde
  $2^{\aleph_0}$, kardinal kuvvet olarak anla\c s\i l\i r,
  ve bu kuvvet, $2^{\upomega}$ ordinal kuvvetinden farkl\i d\i r.
  Asl\i nda $2^{\upomega}=\upomega$, ama sonraki teoreme g\"ore
  $2^{\aleph_0}>\aleph_0$.
\end{sloppypar}

\begin{theorem}
  $2^{\kappa}=\card{\pow{\kappa}}$.
\end{theorem}

\ktk

A\c sa\u g\i daki kurallar kolayd\i r.
\begin{align*}
&\begin{gathered}
	0<\lambda\lto 0^{\lambda}=0,\\
	\kappa^0=1,\\
	1^{\lambda}=1,\\
	\kappa^1=\kappa,
	\end{gathered}&
&\begin{gathered}
	\kappa^{\lambda\cardsum\mu}=\kappa^{\lambda}\cardprod\kappa^{\mu},\\
	\kappa^{\lambda\cardprod\mu}=(\kappa^{\lambda})^{\mu},\\
	\kappa\leq\mu\land\lambda\leq\nu\lto\kappa^{\lambda}\leq\mu^{\nu}.	
\end{gathered}
\end{align*}

\begin{theorem}
$2\leq\kappa$, $1\leq\lambda$, ve $\aleph_0\leq\max\{\kappa,\lambda\}$ olsun.  
O zaman
\begin{gather}\label{eqn:kl}
\kappa\leq 2^{\lambda}\lto\kappa^{\lambda}=2^{\lambda},\\\label{eqn:lk}
\lambda\leq\kappa\lto
%\kappa\leq
\kappa^{\lambda}\leq2^{\kappa}.
\end{gather}
\end{theorem}

\begin{proof}
Hipoteze g\"ore $\kappa\leq2^{\lambda}$ ise 
$2\leq\kappa\leq 2^{\lambda}$ ve $\lambda$ sonsuzdur, dolay\i s\i yla
\begin{equation*}
2^{\lambda}\leq\kappa^{\lambda}\leq(2^{\lambda})^{\lambda}
=2^{\lambda\cardprod\lambda}=2^{\lambda}.
\end{equation*}
Ayr\i ca $\lambda\leq\kappa$ ise $\kappa$ sonsuzdur, dolay\i s\i yla
\begin{equation*}
\kappa\leq\kappa^{\lambda}
\leq(2^{\kappa})^{\lambda}
=2^{\kappa\cardprod\lambda}
=2^{\kappa}.\qedhere
\end{equation*}
\end{proof}

\begin{sloppypar}
  Tekrar $\kappa$ ve $\lambda$'n\ in biri sonsuz olsun.
  E\u ger $\lambda\leq\kappa\leq2^{\lambda}$ ise, 
  o zaman \eqref{eqn:kl} gerektirmesine g\"ore
  \begin{equation*}
    \kappa^{\lambda}=2^{\lambda}\leq2^{\kappa};
  \end{equation*}
  burada \eqref{eqn:lk} gerekmez.
  Bir durumda, 
  e\u ger \eqref{eqn:kl} gerektirmesinin hipotezi do\u gru de\u gilse,
  o zaman $2^{\lambda}<\kappa$, dolay\i s\i yla $\lambda\leq\kappa$,
  ve \eqref{eqn:lk} kullan\i labilir.
  Bu \c sekilde teoremin yerine
  \begin{gather*}%\label{eqn:kl}
    \kappa\leq 2^{\lambda}\lto\kappa^{\lambda}=2^{\lambda},\\%\label{eqn:lk}
    2^{\lambda}<\kappa\lto
    %\kappa\leq
    \kappa^{\lambda}\leq2^{\kappa}.
  \end{gather*}
  kurallar\i\ kullan\i labilir.
  (Tekrar
  $2\leq\kappa$, $1\leq\lambda$, 
  ve $\aleph_0\leq\max\{\kappa,\lambda\}$ olmal\i d\i r.)
  \"Orne\u gin
  \begin{gather*}
    2\leq\kappa\leq2^{\aleph_0}\lto\kappa^{\aleph_0}=2^{\aleph_0},\\
    2^{\aleph_0}<\kappa\lto%\kappa\leq
    \kappa^{\aleph_0}\leq2^{\kappa}.
  \end{gather*}
\end{sloppypar}

\c Simdi a\c sa\u g\i daki tan\i m yapabiliriz.
\begin{gather*}
  \beth_0=\aleph_0,\\
\beth_{\alpha'}=\card{\pow{\beth_{\alpha}}}=2^{\beth_{\alpha}},\\
\alpha\text{ limit ise }\beth_{\alpha}=\sup_{\xi<\alpha}\beth_{\xi}.
\end{gather*}
(Burada $\beth$, \.Ibrani \emph{beth} harfidir.)
O zaman $\xi\mapsto\beth_{\xi}$ normaldir, ve
\begin{equation*}
  \aleph_{\alpha}\leq\beth_{\alpha}.
\end{equation*}

\begin{theorem}
T\"um $\kappa$ ve $\lambda$ i\c cin
\begin{gather*}
2\leq\kappa\leq\beth_{\alpha+1}\lto\kappa^{\beth_{\alpha}}=\beth_{\alpha+1},\\
1\leq\lambda\leq\beth_{\alpha}\lto{\beth_{\alpha+1}}^{\lambda}=\beth_{\alpha+1}.
\end{gather*}
\end{theorem}

\ktk

\textbf{Kontinuum Hipotezi} veya \textbf{KH,}
$\aleph_1=\beth_1$ \"onermesidir.
\textbf{Genelle\c stirilmi\c s Kontinuum Hipotezi} veya \textbf{GKH,}
$\Forall{\xi}\aleph_{\xi}=\beth_{\xi}$ \"onermesidir.
G\"odel'in kan\i tlad\i\u g\i\ teoreme g\"ore, 
ZFC aksiyomlar\i n\i n bir modelinde,
GKH do\u grudur.
Cohen'in kan\i tlad\i\u g\i\ teoreme g\"ore, 
ZFC aksiyomlar\i n\i n bir modelinde,
KH yanl\i\c st\i r.
Bu \c sekilde KH, ZFC'den ba\u g\i ms\i zd\i r.

\appendix

\chapter{Harfler}\label{ch:let}

Metinde simge olarak kullan\i l\i rken harfler a\c sa\u g\i daki anlamlara gelir.

\begin{description}
\item[``Tahta siyah\i'' harfleri]\mbox{}
\begin{itemize}[]
\item
$\R$\letsep ger\c cel say\i lar k\"umesi \letterrefs{\sayfada{R}}
\item
$\Q$\letsep kesirli say\i lar k\"umesi \letterrefs{\sayfada{Q}}
\item
$\Z$\letsep tamsay\i lar k\"umesi \letterrefs{\sayfada{Z}}
\item
$\N$\letsep $\{1,2,3,\dots\}$ sayma say\i lar k\"umesi \letterrefs{\sayfada{N}}
\end{itemize}
\item[K\"u\c c\"uk Latin harfleri]\mbox{}
\begin{itemize}[]
\item
$a$, $b$, $c$, $d$, $e$\letsep say\i lar veya k\"umeler
\item
$f$, $g$, $h$\letsep k\"umede tan\i mlanmi\c s g\"ondermeler
\item
$i$, $j$\letsep do\u gal say\i\ de\u gi\c skenler
\item
$k$, $\ell$, $m$, $n$\letsep do\u gal say\i lar
\item
$p$\letsep asal say\i\ \letterrefs{\ref{p} ve \sayfada{p2}}
\item
$u$, $x$, $y$, $z$\letsep say\i\ veya k\"ume de\u gi\c skenleri
\end{itemize}
\item[Dikey k\"u\c c\"uk Latin harfleri]\mbox{}
\begin{itemize}[]
\item
$\sup$\letsep supremum
\item
$\min$\letsep minimum (en k\"u\c c\"uk)
\item
$\max$\letsep maksimum (en b\"uy\"uk)
\end{itemize}
\item[B\"uy\"uk Latin harfleri]\mbox{}
\begin{itemize}[]
\item
$A$, $B$, $C$, $D$\letsep k\"umeler \letterrefs{\sayfada{A}}
\item
$X$, $Y$, $Z$\letsep k\"ume de\u gi\c skenleri
\end{itemize}
\item[K\i v\i rc\i k Latin harfleri]\mbox{}
\begin{itemize}[]
\item
$\mathscr A$, $\mathscr B$, $\mathscr C$\letsep elemanlar\i\ k\"ume veya g\"onderme olan k\"umeler
\letterrefs{\pageref{curly-C} ve \sayfanumarada{curly-B}}
\item
$\pow A$\letsep$\{X\colon X\included A\}$ \letterrefs{\pageref{pow} ve \sayfanumarada{thm:pow}}
\item
$\powf A$\letsep$\{X\in\pow A\colon\card X<\upomega\}$ \letterrefs{\sayfada{powf}}
\end{itemize}
\item[B\"uy\"uk siyah Latin harfleri]\mbox{}
\begin{itemize}[]
\item
$\bm A$, $\bm B$, $\bm C$\letsep s\i n\i flar \letterrefs{\S \ref{sect:sets-classes}, \sayfada{sect:sets-classes}}
\item
$\bm F$, $\bm G$, $\bm H$\letsep s\i n\i fta tan\i mlanm\i\c s g\"ondermeler
\letterrefs{\Teorem{thm:ord-rec}, \sayfada{thm:ord-rec}}
\end{itemize}
\item[Dikey b\"uy\"uk siyah Latin harfleri]\mbox{}
\begin{itemize}[]
\item
$\universe$\letsep evrensel s\i n\i f
\item
$\on$\letsep ordinaller s\i n\i f\i
\item
$\cn$\letsep kardinaller s\i n\i f\i
\end{itemize}
\item[Yunan harfleri]\mbox{}
\begin{itemize}[]
\item
$\alpha$, $\beta$, $\gamma$, $\delta$, $\theta$\letsep ordinaller \letterrefs{\sayfada{minus-gr}}
\item
$\xi$, $\eta$, $\zeta$\letsep ordinal de\u gi\c skenler \letterrefs{\sayfada{minus-gr}}
\item
$\kappa$, $\lambda$, $\mu$, $\nu$\letsep kardinaller \letterrefs{\sayfada{card}}
\item
$\phi$, $\psi$, $\chi$\letsep form\"uller \letterrefs{\sayfada{phi}}
\end{itemize}
\item[Dikey Yunan harfi]\mbox{}
\begin{itemize}[]
\item
$\upvarepsilon_0$\letsep$\sup\{\upomega,\upomega^{\upomega},\upomega^{\upomega^{\upomega}},\dots\}$
\letterrefs{\sayfada{epsilon}}
\item
$\upomega$\letsep $\{0,1,2,\dots\}$ do\u gal say\i lar\i\ k\"umesi \letterrefs{\sayfada{eqn:omega=}}
\end{itemize}
\item[Harflerden t\"ureyen simgeler]\mbox{}
\begin{itemize}[]
\item
$\in$\letsep eleman olma ba\u g\i nt\i s\i\ 
(``$a$ \gr{>est`i} $B$'' demek ``$a$, bir $B$'dir'')
\item
$\forall$\letsep her \lips i\c cin (\emph{for \textbf All})
\item
$\exists$\letsep baz\i\lips i\c cin (\emph{there \textbf Exists})
\item
$\cup$, $\bigcup$\letsep bile\c sim (\emph{\textbf Union})
\end{itemize}
\end{description}

\chapter{Mant\i k}\label{mantik}

\section{Form\"uller}

Form\"ullerde kulland\i\u g\i m\i z simgelerin birka\c c tane t\"ur\"u vard\i r:
\begin{compactenum}[1)]
\item
\textbf{de\u gi\c skenler}\index{de\u gi\c sken} (\eng{variables}):
$z$, $y$, $x$, \dots; $x_0$, $x_1$, $x_2$, \dots;%
\glossary{$x$, $y$, $z$, \dots}
\item
\textbf{sabitler}\index{sabit} (\eng{constants}): $a$, $b$, $c$,
\dots; $a_0$, $a_1$, $a_2$, \dots;%
\glossary{$a$, $b$, $c$, \dots}
\item
\textbf{iki-konumlu ba\u glay\i c\i lar}\index{ba\u glay\i c\i} (\eng{binary connectives}): $\land$, $\lor$, $\lto$, $\liff$;\footnote{Bazen $\lto$ ile $\liff$ oklar\i n\i n yerine $\to$ ile $\leftrightarrow$ i\c saretleri yaz\i l\i r.}
\item
bir \textbf{tek-konumlu ba\u glay\i c\i} (\eng{singulary connective}): $\lnot$;
\item
\textbf{niceleyiciler}\index{niceleyici} (\eng{quantifiers}): $\exists$, $\forall$;
\item
\textbf{ayra\c clar}\index{ayra\c c} (\eng{parentheses, brackets}): $($, $)$;
\item
bir \textbf{y\"uklem}\index{y\"uklem} (\eng{predicate}): $\in$ (epsilon).
\end{compactenum}
Bir \textbf{terim}%
\index{terim}\label{terim}
(\eng{term}), ya de\u gi\c sken ya da sabittir.  E\u ger $t$ ile $u$, iki terim ise, o zaman
\begin{equation*}
t\in u
\end{equation*}
ifadesi, bir \textbf{b\"ol\"unemeyen form\"uld\"ur}%
\index{form\"ul} (\eng{atomic formula}).  
Genelde \textbf{form\"ullerin} tan\i m\i, \"ozyinelidir:
\begin{compactenum}
\item
B\"ol\"unemeyen bir form\"ul, bir form\"uld\"ur.
\item
E\u ger $\phi$ bir form\"ul ise, o zaman
\begin{equation*}
\lnot\phi
\end{equation*}
ifadesi de bir form\"uld\"ur.
\item
E\u ger $\phi$ ile $\psi$ iki form\"ul ise, o zaman
\begin{align*}
&(\phi\land\psi),&
&(\phi\lor\psi),&
&(\phi\lto\psi),&
&(\phi\liff\psi)
\end{align*}
 ifadeleri de form\"uld\"ur.
\item
E\u ger $\phi$ bir form\"ul ise, ve $x$ bir de\u gi\c sken ise, o zaman
\begin{align*}
&\Exists x\phi,&\Forall x\phi
\end{align*}
ifadeleri de form\"uld\"ur.
\end{compactenum}
Form\"ullerin her t\"ur\"un\"un ad\i\ vard\i r:
\begin{compactenum}
\item
$\lnot\phi$ form\"ul\"u, bir 
\textbf{de\u gillemedir}%
\index{de\u gilleme}
(\eng{negation}).
\item
$(\phi\land\psi)$ form\"ul\"u, bir 
\textbf{birle\c sme}%
\index{birle\c sme}
veya \textbf{t\"umel evetlemedir}
\index{t\"umel evetleme}%
\index{evetleme}
(\eng{conjunction}).
\item
$(\phi\lor\psi)$ form\"ul\"u, bir 
\textbf{ayr\i lma}%
\index{ayr\i lma}
veya \textbf{tikel evetlemedir} 
(\eng{disjunction}). 
\item
$(\phi\lto\psi)$ form\"ul\"u, bir 
\textbf{gerektirme}%
%\textbf{kar\i\c st\i rmad\i r}
\index{gerektirme}
(\eng{implication}). 
\item
$(\phi\liff\psi)$ form\"ul\"u, bir 
\textbf{denkliktir}%
\index{denklik}
(\eng{equivalence}). 
\item
$\Exists x\phi$ form\"ul\"u, bir 
\textbf{\"orneklemedir}%
\index{\"ornekleme}
(\eng{instantiation}). 
\item
$\Forall x\phi$ form\"ul\"u, bir 
\textbf{genelle\c stirmedir}%
\index{genelle\c stirme}
(\eng{generalization}). 
\end{compactenum}
Bu t\"urlerin adlar\i, \c cok \"onemli de\u gildir.  Fakat a\c sa\u
g\i daki teorem \c cok \"onemlidir. 

\begin{theorem}
Her form\"ul\"un tek bir \c sekilde tek bir t\"ur\"u vard\i r.
\end{theorem}

Mesela ayn\i\ form\"ul, hem gerektirme, hem \"ornekleme olamaz:
$\Exists x(\phi\lto\psi)$ form\"ul\"u, gerektirme de\u gil,
\"orneklemedir; $(\Exists x\phi\lto\psi)$ form\"ul\"u, \"ornekleme de\u gil, gerektirmedir.

Ayr\i ca $(\phi\land(\psi\land\theta))$ form\"ul\"u, tek bir \c
sekilde birle\c smedir.  Asl\i nda sadece $\phi$ ile
$(\psi\land\theta)$ form\"ullerinin birle\c smesidir.  E\u ger $A$
harf\/i, $\phi\land(\psi$ ifadesini g\"osterirse ve $B$ harf\/i,
$\theta)$ ifadesini g\"osterirse, o zaman $(A\land B)$ ifadesi,
$(\phi\land(\psi\land\theta))$ form\"ul\"un\"u g\"osterir; ama tan\i
ma g\"ore bu form\"ul, $A$ ile $B$ ifadelerinin birle\c smesi de\u
gildir, \c c\"unk\"u $A$ ile $B$ ifadeleri (yani $A$ ile $B$ taraf\i
ndan g\"osterilen ifadeler), form\"ul de\u gildir. 

Teoremi kan\i tlamayaca\u g\i z.  
Fakat teoremi kullanarak a\c sa\u g\i daki \"ozyineli tan\i m\i\ yapabiliriz.  
Bir de\u g\i\c skenin bir form\"ulde birka\c c tane 
\textbf{ge\c ci\c si}%
\index{ge\c cis}
(\eng{occurrence})
olabilir.  Mesela $\Forall x(x\in y\liff x\in z)$ form\"ul\"unde $x$ de\u gi\c skeninin \"u\c c tane ge\c ci\c si vard\i r (ve $y$ ile $z$ de\u gi\c skenlerinin birer ge\c ci\c si vard\i r).
\begin{compactenum}
\item
B\"ol\"unemeyen bir form\"ulde bir de\u gi\c skenin her ge\c ci\c si,
\textbf{serbest} bir ge\c ci\c stir.
\item
Bir de\u gi\c skenin $\phi$ form\"ul\"undeki her serbest ge\c ci\c si,
$\lnot\phi$, $(\phi*\psi)$, ve $(\psi*\phi)$ form\"ullerinde de
serbesttir.  (Burada $*$ i\c sareti, herhangi bir iki-konumlu ba\u glay\i c\i
d\i r.) 
\item
E\u ger $x$ ile $y$, iki \emph{farkl\i} de\u gi\c sken ise, o zaman $x$ de\u
gi\c skeninin $\phi$ form\"ul\"unde her serbest ge\c ci\c si, $\Exists
y\phi$ ile $\Forall y\phi$ form\"ullerinde de serbesttir. 
\item
$\Exists x\phi$ ile $\Forall x\phi$ form\"ullerinde $x$ de\u gi\c
  skeninin hi\c c serbest ge\c ci\c si yoktur.
\end{compactenum}
Bir form\"ulde bir de\u gi\c skenin serbest ge\c ci\c si varsa, bu
de\u gi\c sken, form\"ul\"un bir \textbf{serbest de\u gi\c skenidir.}
Serbest de\u gi\c skeni olmayan bir form\"ul, bir
\textbf{c\"umledir.}\index{c\"umle}  C\"umleler i\c cin $\sigma$, $\tau$,
ve $\rho$ gibi Yunan harflerini kullanaca\u g\i z. 

\section{Do\u gruluk ve yanl\i\c sl\i k}

Bir $\phi$ form\"ul\"un\"un tek serbest de\u gi\c skeni $x$ ise, o
zaman form\"ul 
\begin{equation*}
\phi(x)
\end{equation*}
olarak yaz\i labilir.  O halde $a$ bir sabit ise, ve $x$ de\u gi\c
skeninin $\phi$ form\"ul\"undeki her \emph{serbest} ge\c ci\c sinin yerine
$a$ konulursa, \c c\i kan c\"umle 
\begin{equation*}
\phi(a)
\end{equation*}
olarak yaz\i labilir.  \c Simdi 
\textbf{do\u grulu\u gu}%
\index{do\u gruluk}\label{truth}
(\eng{truth}) ve
\textbf{yanl\i\c sl\i\u g\i}%
\index{yanl\i\c sl\i k}
(\eng{falsehood})
tan\i mlayabiliriz: 
\begin{compactenum}
\item
E\u ger $b$ k\"umesi, $a$ k\"umesini i\c cerirse, o zaman $a\in b$
c\"umlesi do\u grudur; i\c cermezse, yanl\i\c st\i r. 
\item
E\u ger $\sigma$ c\"umlesi do\u gruysa, o zaman $\lnot\sigma$ de\u
gillemesi yanl\i\c st\i r; $\sigma$ yanl\i\c s ise, $\lnot\sigma$ do\u
grudur. 
\item
E\u ger hem $\sigma$ hem $\tau$ do\u gruysa, o zaman
$(\sigma\land\tau)$ birle\c smesi de do\u grudur; $\sigma$ ile $\tau$
c\"umlelerinin biri yanl\i\c s ise, birle\c smesi de yanl\i\c st\i r. 
\item
E\u ger bir $a$ k\"umesi i\c cin $\phi(a)$ c\"umlesi do\u gruysa, o
zaman $\Exists x\phi(x)$ \"orneklemesi de do\u grudur; hi\c c \"oyle
bir $a$ yoksa, \"ornekleme yanl\i\c st\i r. 
\item
$(\sigma\lor\tau)$ c\"umlesi, $\lnot(\lnot\sigma\land\lnot\tau)$
  c\"umlesinin anlam\i na gelir, yani bu iki c\"umle ayn\i\ zamanda ya
  do\u grudur, ya da yanl\i\c st\i r. 
\item
$(\sigma\lto\tau)$ c\"umlesi, $(\lnot\sigma\lor\tau)$ c\"umlesinin anlam\i na gelir.
\item
$(\sigma\liff\tau)$ c\"umlesi, $\bigl((\sigma\lto\tau)\land(\tau\lto\sigma)\bigr)$ c\"umlesinin anlam\i na gelir.
\item
$\Forall x\phi(x)$ c\"umlesi, $\lnot\Exists x\lnot\phi(x)$ c\"umlesinin anlam\i na gelir.
\end{compactenum}
\"Ozel olarak form\"ullerde $\lor$, $\lto$, $\liff$, ve $\forall$
simgeleri gerekmez; sadece kolayl\i k i\c cin kullanaca\u g\i z.  Ama
$(\sigma\lto\tau)$ c\"umlesi do\u grudur ancak ve ancak $\tau$ do\u
gru veya $\sigma$ yanl\i\c st\i r; ve $(\sigma\liff\tau)$ c\"umlesi
do\u grudur ancak ve ancak hem $\sigma$ hem $\tau$ ya do\u gru ya
yanl\i\c st\i r.  Ayr\i ca $\Forall x\phi(x)$ do\u grudur ancak ve
ancak her $a$ k\"umesi i\c cin $\phi(a)$ do\u grudur. 

Birka\c c tane k\i saltma daha kullan\i r\i z:
\begin{compactenum}
\item
$\lnot\; t\in u$ form\"ul\"un\"un yerine $t\notin u$ ifadesini yazar\i z;
\item
Bir $(\phi*\psi)$ form\"ul\"un\"un en d\i\c staki ayra\c clar\i
n\i\ yazmay\i z.
\item
$\lto$ ile $\liff$ ba\u glay\i c\i lar\i na g\"ore $\land$ ile
$\lor$ ba\u glay\i c\i lar\i na \"onceli\u gi veririz:  Mesela
$\phi\land\psi\lto\chi$ ifadesi, $(\phi\land\psi)\lto\chi$
form\"ul\"un\"un anlam\i na gelir.   
\item
$\phi\lto\psi\lto\chi$ ifadesi, $\phi\lto(\psi\lto\chi)$
form\"ul\"un\"un anlam\i na gelir. 
\end{compactenum}
Bir $\phi$ form\"ul\"un\"un serbest de\u gi\c skenleri $x$ ile $y$
ise, o zaman form\"ul 
\begin{equation*}
\phi(x,y)
\end{equation*}
olarak yaz\i labilir.  O halde $a$ ile $b$, iki sabit ise, ve $x$ de\u
gi\c skeninin $\phi$ form\"ul\"undeki her serbest ge\c ci\c sinin
yerine $a$ konulursa, ve benzer \c sekilde $y$ de\u gi\c skeninin her
serbest ge\c ci\c sinin yerine $b$ konulursa, \c c\i kan c\"umle
\begin{equation*}
\phi(a,b)
\end{equation*}
olarak yaz\i labilir.  

Genelde $\phi$ form\"ul\"un\"un serbest de\u gi\c skenleri, bir $\vec
x$ listesini olu\c sturursa, o zaman form\"ul 
\begin{equation*}
\phi(\vec x)
\end{equation*}
olarak yaz\i labilir; ayr\i ca
\begin{align*}
\Forall{\vec x}&\phi(\vec x),&
\Exists{\vec x}&\phi(\vec x)
\end{align*}
c\"umleleri yaz\i labilir.  E\u ger $\vec a$, uzunlu\u gun $\vec x$
listesinin uzunlu\u gu olan bir sabit listesiyse, o zaman 
\begin{equation*}
\phi(\vec a)
\end{equation*}
c\"umlesi de \c c\i kar.
E\u ger $\phi(\vec x)$ ile $\psi(\vec x)$, iki form\"ul ise, ve \emph{sadece
do\u grulu\u gun tan\i m\i n\i\ kullanarak}
\begin{equation*}
\Forall{\vec x}\bigl(\phi(\vec x)\liff\psi(\vec x)\bigr)
\end{equation*}
c\"umlesinin do\u grulu\u gu kan\i tlanabilirse, o zaman $\phi$ ile
$\psi$ birbirine 
\textbf{(mant\i\u ga g\"ore) denktir}\index{denk} (\eng{logically
  equivalent}):
k\i saca
\begin{equation*}
  \phi\denk\psi.
\end{equation*}
\"Oyleyse $\phi$ ile $\psi$ birbirine denktir, ancak
ve ancak her $\vec a$ sabit listesi i\c cin, \emph{do\u grulu\u gun tan\i
m\i na g\"ore}
\begin{equation*}
\phi(\vec a)\liff\psi(\vec a)
\end{equation*}
c\"umlesi do\u grudur.  \"Orne\u gin, yukar\i daki tan\i mlara g\"ore
\begin{gather*}
	\phi\lor\psi\denk\lnot(\lnot\phi\land\lnot\psi),\\
	\phi\lto\psi\denk\lnot\phi\lor\psi,\\
	\phi\liff\psi\denk(\phi\lto\psi)\land(\psi\lto\phi),\\
	\Forall x\phi\denk\lnot\Exists x\lnot\phi.
\end{gather*}
Ama $\Exists y\Forall x\bigl(\phi(x)\lto x\in y\bigr)$ ile
$\Exists y\Forall x\bigl(\phi(x)\liff x\in y\bigr)$, denk de\u gildir.

\begin{theorem}\label{thm:denklik}
\mbox{}
\begin{compactenum}
\item
Her form\"ul, kendisine denktir.
\item
E\u ger $\phi$ ile $\psi$ denk ise, o zaman $\psi$ ile $\phi$ denktir.
\item
E\u ger $\phi$ ile $\psi$ denk ise, ve $\psi$ ile $\chi$ denk ise, o zaman $\phi$ ile $\chi$ denktir.
\end{compactenum}
\begin{comment}
  

Yani
\begin{gather*}
	\phi\denk\phi,\\
	\phi\denk\psi\lto\psi\denk\phi,\\
	\phi\denk\psi\land\psi\denk\chi\lto\phi\denk\chi.
\end{gather*}



\end{comment}
\end{theorem}

\begin{proof}
\begin{asparaenum}
\item
$\sigma\liff\sigma$ her zaman do\u grudur.
\item
$\sigma\liff\tau$ do\u gru olsun.  O zaman hem $\sigma$ hem $\tau$ ya do\u gru ya yanl\i\c st\i r.  \"Oyleyse hem $\tau$ hem $\sigma$ ya do\u gru ya yanl\i\c st\i r; yani $\tau\liff\sigma$ do\u grudur.
\item
$\sigma\liff\tau$ ve $\tau\liff\rho$ do\u gru olsun.  E\u ger $\sigma$ do\u gruysa, o zaman $\tau$ do\u gru olmal\i, ve sonu\c c olarak $\rho$ do\u gru olmal\i, dolay\i s\i yla $\sigma\liff\rho$ do\u grudur.  Benzer \c sekilde $\sigma$ yanl\i\c s ise $\sigma\liff\rho$ tekrar do\u grudur.\qedhere
\end{asparaenum}
\end{proof}

\begin{theorem}\mbox{}\label{thm:lto}
\begin{compactenum}
\item
$\phi\lto\psi\lto\chi$ ile $\phi\land\psi\lto\chi$ denktir.
\item
E\u ger $x$ de\u gi\c skeni, $\phi$ form\"ul\"unde serbest de\u gilse, 
o zaman
\begin{equation*}
\Forall x(\phi\lto\psi)\denk\phi\lto\Forall x\psi.
\end{equation*}
\end{compactenum}
\end{theorem}

\begin{proof}
\begin{asparaenum}
\item
$\sigma\lto\tau\lto\rho$ do\u gru olsun.  E\u ger $\sigma\land\tau$
  c\"umlesi de do\u gruysa, o zaman hem $\sigma$ hem $\tau$ do\u
  grudur, ve sonu\c c olarak $\tau\lto\rho$ do\u grudur, ve $\rho$
  do\u grudur.  Yani $\sigma\land\tau\lto\rho$ do\u grudur. 

Tersi i\c cin $\sigma\land\tau\lto\rho$ do\u gru olsun.  O zaman
$\sigma\land\tau$ yanl\i\c s veya $\rho$ do\u grudur.  Yani $\sigma$
yanl\i\c s, veya $\tau$ yanl\i\c s, veya $\rho$ do\u grudur.  E\u ger
$\sigma$ do\u gruysa, o zaman $\tau$ yanl\i\c s, veya $\rho$ do\u
grudur, yani $\tau\lto\rho$ do\u grudur.  Sonu\c c olarak
$\sigma\lto\tau\lto\rho$ do\u grudur. 

\item
$\Forall x(\sigma\lto\phi(x))$ do\u gru olsun.  
O zaman her $a$ i\c cin $\sigma\lto\phi(a)$ do\u grudur.  
Sonu\c c olarak $\sigma$ do\u gruysa, 
o zaman her $a$ i\c cin $\phi(a)$ do\u gru\-dur.  
Yani $\sigma\lto\Forall x\phi(x)$ do\u grudur.

Benzer \c sekilde $\sigma\lto\Forall x\phi(x)$ do\u gruysa 
$\Forall x(\sigma\lto\phi(x))$ do\u grudur.\qedhere
\end{asparaenum}
\end{proof}

\chapter{Kof\/inallik}\label{ch:cof}

\section{Tan\i m ve \"ozellikler}

\begin{sloppypar}
  Sonsuz bir $\kappa$ kardinali limit ordinali oldu\u gundan
  \begin{equation*}
    \kappa=\sup\{\xi\colon\xi<\kappa\}=\bigcup_{\xi<\kappa}\xi.
  \end{equation*}
  Bazen bir kardinal, 
  kendisinden k\"u\c c\"uk bir altk\"umenin supremumu\-dur.  
  \"Orne\u gin $\upomega<\aleph_{\upomega}$, ama
  \begin{equation*}
    \aleph_{\upomega}=\sup\{\aleph_x\colon x\in\upomega\}.
  \end{equation*}
  Genelde $\alpha$ limit, $b\included\alpha$, ve
  \begin{equation*}
    \Forall{\xi}\bigl(\xi<\alpha\lto\Exists{\eta}(\eta\in b\land\xi<\eta)\bigr)
  \end{equation*}
  ise,
  $b$ altk\"umesi, $\alpha$ ordinalinin \textbf{s\i n\i rs\i z}%
  \index{s\i n\i rs\i z}
  (\eng{unbounded})
  altk\"umesidir.
  Bu durumda
  \begin{equation*}
    \alpha=\sup(b).
  \end{equation*}
  \"Orne\u gin her limit ordinali, kendisinde s\i n\i rs\i zd\i r.  
  Ayr\i ca $\{\aleph_x\colon x\in\upomega\}$, 
  $\aleph_{\upomega}$ ordinalinde s\i n\i rs\i zd\i r.
  Bir limit ordinalinin s\i n\i rs\i z alt\-k\"umelerinin 
  en k\"u\c c\"uk kardinaline, 
  ordinalin
  \textbf{kof\/inalli\u gi}%
  \index{kof{}inallik}
  (\eng{cofinality})
  denir, ve bu kardinal,
  $\cof{\alpha}$
  \glossary{$\cof{\alpha}$}%
  olarak yaz\i labilir.  Yani
  \begin{equation*}
    \cof{\alpha}=\min\{\cardinal(x)\colon x\included\alpha\land\sup(x)=\alpha\}.
  \end{equation*}
  Ayr\i ca, tan\i ma g\"ore,
  \begin{align*}
    \cof0&=0,&\cof{\alpha+1}&=1
  \end{align*}
  denebilir,
  ama bu durumlar\i\ kullanmayaca\u g\i z.
\end{sloppypar}

\begin{theorem}
Her $\alpha$ limit ordinali i\c cin,
tan\i m k\"umesi $\cof{\alpha}$ olan,
de\u ger k\"umesi $\alpha$ ordinalinin s\i n\i rs\i z bir altk\"umesi olan,
kesin artan bir g\"onderme vard\i r.
\end{theorem}

\begin{proof}
$f\colon\cof{\alpha}\to\alpha$ olsun, 
ve $f[\alpha]$, $\alpha$ ordinalinin s\i n\i rs\i z bir altk\"umesi olsun.  
\"Ozyinelemeyle, tan\i m k\"umesi $\cof{\alpha}$ olan,
\begin{equation*}
g(\beta)=\max\Bigl(f(\beta),\sup\bigl(g[\beta]\bigr)\Bigr)
\end{equation*}
ko\c sulunu sa\u glayan bir $g$ g\"ondermesi vard\i r.
E\u ger $\beta<\cof{\alpha}$ ve $g[\beta]\included\alpha$ ise, 
o zaman $g[\beta]$, $\alpha$ ordinalinin s\i n\i rs\i z altk\"umesi de\u gil,
dolay\i s\i yla $g(\beta)\in\alpha$; ayr\i ca $f(\beta)\leq g(\beta)$.  
\"Oyleyse $g$, istedi\u gimiz gibidir.
\end{proof}

\begin{theorem}
$\alpha$ ve $\beta$ limit ordinalleri olsun.
E\u ger $f\colon\alpha\to\beta$ ve kesin artan ise,
ve $\beta=\bigcup f[\alpha]$ ise,
o zaman
\begin{equation*}
\cof{\alpha}=\cof{\beta}.
\end{equation*}
\end{theorem}

\begin{proof}
$\cof{\beta}\leq\cof{\alpha}$ ve $\cof{\alpha}\leq\cof{\beta}$
e\c sitsizliklerini kan\i tlayaca\u g\i z.
\begin{asparaenum}
\item
$g\colon\cof{\alpha}\to\alpha$
ve $\bigcup g[\cof{\alpha}]=\alpha$ olsun.  
%$(f\circ g)[\gamma]$ g\"or\"unt\"us\"un\"un $\beta$ ordinalinde s\i n\i rs\i z oldu\u gunu kan\i tlayaca\u g\i z.  
$\delta<\beta$ ise,
hipoteze g\"ore $\alpha$ ordinalinin bir $\theta$ eleman\i\ i\c cin
\begin{equation*}
\delta<f(\theta).
\end{equation*}
O zaman $\cof{\alpha}$ kardinalinin bir $\iota$ eleman\i\ i\c cin
\begin{align*}
\theta&<g(\iota),&
\delta<f(\theta)&<f\bigl(g(\iota)\bigr).
\end{align*}
\"Oyleyse $\bigcup(f\circ g)[\cof{\alpha}]=\beta$,
dolay\i s\i yla $\cof{\beta}\leq\cof{\alpha}$.
\item
$h\colon\cof{\beta}\to\beta$ ve $\bigcup h[\cof{\beta}]=\beta$ olsun.
$\delta<\cof{\beta}$ ise
\begin{equation*}
k(\delta)=\min\{\xi\in\alpha\colon h(\delta)<f(\xi)\}
\end{equation*}
olsun.  O zaman $k\colon\cof{\beta}\to\alpha$.  
E\u ger $\theta\in\alpha$ ise, o zaman
$\cof{\beta}$ kardinalinin
\begin{equation*}
  f(\theta)<h(\delta)
\end{equation*}
ko\c sulunu sa\u glayan bir $\delta$ eleman\i\ vard\i r.
O zaman
\begin{equation*}
f(\theta)<h(\delta)<f(k(\delta)),  
\end{equation*}\sloppy
dolay\i s\i yla $\theta<k(\delta)$,
\c c\"unk\"u $f$ kesin artand\i r.
\"Oyleyse $\bigcup k[\cof{\beta}]=\alpha$,
dolay\i s\i yla
$\cof{\alpha}\leq\cof{\beta}$ 
ve asl\i nda $\cof{\alpha}=\cof{\beta}$.\qedhere
\end{asparaenum}
\end{proof}

\"Ozel durum olarak $\bm F$ normal ve $\alpha$ limit ise
\begin{equation*}
  \cof{\bm F(\alpha)}=\cof{\alpha}.
\end{equation*}

\begin{theorem}
  $\alpha$ limit ise $\cof{\aleph_{\alpha}}=\cof{\alpha}$.
\end{theorem}

\begin{proof}
$\xi\mapsto\aleph_{\xi}$ normaldir.
\end{proof}

\begin{theorem}
  Cantor normal bi\c ciminde
\begin{equation*}
\alpha=\upomega^{\alpha_0}\cdot a_0+\dots+\upomega^{\alpha_n}\cdot a_n
\end{equation*}
ve $\alpha_n>0$ ise, o zaman
\begin{equation*}
\cof{\alpha}
=\begin{cases}
	\upomega,&\text{ e\u ger $\alpha_n$ bir ard\i lsa},\\
	\cof{\alpha_n},&\text{ e\u ger $\alpha_n$ bir limitse}.
\end{cases}
\end{equation*}
\end{theorem}

\begin{proof}
Son teoreme g\"ore
$\alpha$ limit, $\gamma\geq1$, ve $\delta\geq2$ ise
\begin{equation*}
  \cof{\alpha} 
=\cof{\beta+\alpha} 
=\cof{\gamma\cdot\alpha}
=\cof{\delta^{\alpha}}.\qedhere
\end{equation*}
\end{proof}


Bazen bu hesaplama bize yard\i m etmez.  Mesela $f(0)=0$ ve $f(n+1)=\upomega^{f(n)}$ ve $\alpha=\sup(f[\upomega])$ ise, yani
\begin{equation*}
\alpha =\sup\{0,1,\upomega,\upomega^{\upomega},\upomega^{\upomega^{\upomega}},\dots\}
\end{equation*}
ise, o zaman $\cof{\alpha}=\upomega$, ama $\alpha=\upomega^{\alpha}$.

\begin{theorem}\label{thm:succ-cof}
Her $\alpha$ ordinali i\c cin
\begin{equation*}
\cof{\aleph_{\alpha+1}}=\aleph_{\alpha+1}.  
\end{equation*}
\end{theorem}

\begin{proof}
$\beta<\aleph_{\alpha+1}$ ve $f\colon\beta\to\aleph_{\alpha+1}$ olsun.  O zaman
\begin{equation*}
\sup(f[\beta])=\bigcup_{\xi<\beta}f(\xi).
\end{equation*}
Bu bile\c simden $\aleph_{\alpha}\times\aleph_{\alpha}$ \c carp\i m\i na giden bir $h$ g\"ommesini tan\i mlayaca\u g\i z.
Se\c cim Aksiyomu sayesinde $\bigcup\{{}^{\xi}\aleph_{\alpha}\colon\xi<\aleph_{\alpha+1}\}$ k\"umesi iyi\-s\i ralanabilir.  Bu s\i ralamaya g\"ore $\delta<\aleph_{\alpha+1}$ ise  ${}^\delta\aleph_{\alpha}$ k\"umesinin en k\"u\c c\"uk \emph{g\"ommesi,} $g_{\delta}$ olsun.  O zaman $\gamma<\sup(f[\beta])$ ise
\begin{align*}
\delta&=\min\{z\in\beta\colon\gamma<f(z)\},& h(\gamma)&=\bigl(g_{\beta}(\delta),g_{\delta}(\gamma)\bigr)
\end{align*}
olsun.
B\"oylece
\begin{equation*}
\cardinal\bigl(\sup(f[\beta])\bigr)\leq\cardinal(\aleph_{\alpha}\times\aleph_{\alpha})=\aleph_{\alpha},
\end{equation*}
dolay\i s\i yla $\sup(f[\beta])<\aleph_{\alpha+1}$.  Sonu\c c olarak $\cof{\aleph_{\alpha+1}}=\aleph_{\alpha+1}$.
\end{proof}

\section{Hesaplamalar}

\begin{theorem}
$2\leq\kappa$, $1\leq\lambda$, ve $\aleph_0\leq\max\{\kappa,\lambda\}$ olsun.  O zaman
\begin{align*}
	\lambda\geq\cof{\kappa}&\lto\kappa<\kappa^{\lambda},\\
	\gch\land\lambda<\cof{\kappa}&\lto\kappa=\kappa^{\lambda}.
\end{align*}
\end{theorem}

\begin{proof}
$\cof{\kappa}\leq\lambda$ ise ${}^{\lambda}\kappa$ k\"umesinin
\begin{equation*}
\kappa=\bigcup_{\xi<\lambda}f(\xi)
\end{equation*}
ko\c sulunu sa\u glayan bir $f$ eleman\i\ vard\i r.  
\c Simdi $\xi\mapsto g_{\xi}\colon\kappa\to{}^{\lambda}\kappa$ olsun.  
O zaman ${}^{\lambda}\kappa$ k\"umesinin 
$\{g_{\xi}\colon\xi<\kappa\}$ k\"umesinde olmayan bir
\begin{equation*}
\eta\mapsto\min\Bigl(\kappa\setminus\bigl\{g_{\xi}(\eta)\colon\xi<f(\eta)\bigr\}\Bigr)
\end{equation*}
eleman\i\ vard\i r.  

\c Simdi $\lambda<\cof{\kappa}$ olsun.  
O zaman \Teoremin{thm:succ-cof} kan\i t\i ndaki gibi
\begin{multline*}
{}^{\lambda}\kappa
=\bigcup_{\xi<\kappa}{}^{\lambda}\xi
=\bigcup_{\lambda\leq\xi<\kappa}{}^{\lambda}\xi\\
\preccurlyeq\bigcup_{\lambda\leq\xi<\kappa}{}^{\lambda}(\card{\xi})
=\bigcup_{\substack{\lambda\leq\xi<\kappa\\\xi\in\cn}}{}^{\lambda}\xi
\preccurlyeq\bigcup_{\substack{\lambda\leq\xi<\kappa\\\xi\in\cn}}{}^{\xi}2.
\end{multline*}
E\u ger $\gch$ do\u gruysa $\mu<\kappa\lto2^{\mu}\leq\kappa$, dolay\i s\i yla $\kappa^{\lambda}\leq\kappa$.
\end{proof}

\c Simdi, g\"osterdiklerimize g\"ore, e\u ger $\kappa+\lambda$
sonsuzsa, o zaman
\begin{align*}
2\leq\kappa\leq2^{\lambda}&\lto\kappa^{\lambda}=2^{\lambda},\\
\cof{\kappa}\leq\lambda\leq\kappa&\lto\kappa<\kappa^{\lambda}\leq2^{\kappa},\\
1\leq\lambda<\cof{\kappa}&\lto\kappa\leq\kappa^{\lambda}\leq2^{\kappa}.
\end{align*}
Ayr\i ca%, e\u ger $\gch$ do\u gruysa,
\begin{equation*}
\gch\lto
\kappa^{\lambda}=
\begin{cases}
  \lambda^+,&\text{ e\u ger $2\leq\kappa<\lambda$ ise},\\
\kappa^+,&\text{ e\u ger $\cof{\kappa}\leq\lambda\leq\kappa$ ise},\\
\kappa,&\text{ e\u ger $1\leq\lambda<\cof{\kappa}$ ise}.
\end{cases}
\end{equation*}
\"Ozel olarak% e\u ger $\gch$ do\u gruysa,
\begin{equation*}
\gch\lto
{\aleph_{\alpha}}^{\aleph_{\beta}}=
\begin{cases}
	\aleph_{\beta+1},&\text{ e\u ger $\alpha<\beta$ ise},\\
	\aleph_{\alpha+1},&\text{ e\u ger $\cof{\alpha}\leq\aleph_{\beta}\leq\aleph_{\alpha}$ ise},\\
	\aleph_{\alpha},&\text{ e\u ger $\aleph_{\beta}<\cof{\alpha}$ ise}.	
\end{cases}
\end{equation*}

  
\end{document}

