\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}

%\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}
\ifoot{\leftmark}
%\cehead{K\"umeler Kuram\i}
%\cohead{\today}

\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}

\newtheorem{theorem}{Teorem}
\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}

\newcommand{\ktk}[1][]{\begin{xca}Teoremi kan\i tlay\i n.#1\end{xca}}

\usepackage[all]{xy}
\usepackage{bm}

\newcommand{\lto}{\Rightarrow}
\newcommand{\liff}{\Leftrightarrow}
\newcommand{\Forall}[1]{\forall{#1}\;}
\newcommand{\Exists}[1]{\exists{#1}\;}
\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}}
\newcommand{\card}{\operatorname{kard}}
%\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{\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{\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}

\begin{document}
%\frontmatter
\title{Ordinal Analiz}
\author{David Pierce}
\date{\today\ tasla\u g\i}
%\date{8 \c Subat 2016}
%\publishers{Matematik B\"ol\"um\"u\\
%Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
%\.Istanbul\\
%\url{dpierce@msgsu.edu.tr}\\
%\url{http://mat.msgsu.edu.tr/~dpierce/}}


\maketitle

\tableofcontents

%\listoffigures

%\mainmatter

\section{Ger\c cel Analiz}

Ger\c cel say\i lar,
tam s\i ral\i\ $\R$
cismini olu\c sturur, yani
\begin{compactenum}[1)]
\item
$<$ ba\u glant\i s\i\ taraf\i ndan $\R$ do\u grusal s\i ralanm\i\c st\i r;
\item
bu s\i ralama \textbf{tamd\i r,}
yani bo\c s olmayan, \"usts\i n\i r\i\ olan $\R$'nin her altk\"umesinin \textbf{supremumu}
(en k\"u\c c\"uk \"usts\i n\i r\i) vard\i r;
\item
toplama ($+$) ve carpma ($\times$ veya ${}\cdot{}$) alt\i nda $\R$ bir cisimdir;
\item
s\i f\i r olmayan her ger\c cel $a$ say\i s\i\ i\c cin
\begin{equation*}
a>0\iff-a<0;
\end{equation*}
\item
her iki pozitif ger\c cel say\i n\i n toplam\i\ ve \c carp\i m\i\ pozitiftir.
\end{compactenum}
$\R$'nin bu \"ozelliklerini,
\c simdilik $\R$'nin \emph{aksiyomlar\i} olarak kabul ediyoruz.
(Sonra, \emph{k\"ume aksiyomlar\i n\i} kullanarak
ger\c cel say\i lar\i\ in\c sa edebilece\u giz.)

$\R$'nin her $A$ altk\"umesi i\c cin
\begin{compactenum}[1)]
\item
$1\in A$ ve
\item
$A$'n\i n her $b$ eleman\i\ i\c cin $b+1\in A$
\end{compactenum}
durumunda $A$'ya \textbf{t\"umevar\i ml\i} densin.
O zaman tan\i m\i\ g\"ore
\begin{equation*}
\N=\bigcap\{X\included\R\colon X\text{ t\"umevar\i ml\i d\i r}\}
\end{equation*}
olsun, yani \textbf{sayma say\i s\i} olmak i\c cin gerek ve yeter ko\c sul,
$\R$'nin her t\"umevar\i ml\i\ altk\"umesinin eleman\i\ olmakt\i r.


\begin{theorem}[T\"umevar\i m]
$\N$ t\"umevar\i ml\i d\i r.
Ayr\i ca $\N$'nin tek t\"umevar\i ml\i\ altk\"umesi, kendisidir.
\end{theorem}

\begin{lemma}\label{lem:1leq}
En k\"u\c c\"uk sayma say\i s\i\ vard\i r,
ve bu say\i\ $1$'dir.
\end{lemma}

\begin{lemma}\label{lem:-}
Her sayma say\i s\i, ya $1$'dir,
ya da bir $k$ sayma say\i s\i\ i\c cin $k+1$'dir.
\end{lemma}

\begin{lemma}\label{lem:k<m}
Herhangi $k$ ve $m$ sayma say\i lar\i\ i\c cin
\begin{equation*}%\label{eqn:k<m}
k\leq m\lto k<m+1.
\end{equation*}
\end{lemma}

\begin{theorem}[G\"u\c clu t\"umevar\i m]
$A\included\N$ olsun,
ve t\"um $k$ sayma say\i lar\i\ i\c cin
\begin{equation*}
\{x\in\N\colon x<k\}\included A\lto k\in A
\end{equation*}
olsun.  O zaman $A=\N$.
\end{theorem}

\begin{theorem}[\.Iyis\i ralama]
$\N$ \textbf{iyis\i ral\i d\i r,} yani $\N$'nin bo\c s olmayan her altk\"umesinin en k\"u\c c\"uk eleman\i\ vard\i r.
\end{theorem}

\begin{theorem}[\"Ozyineleme]
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 $\N$'den $A$'ya giden bir ve tek bir $g$ g\"ondermesi i\c cin
\begin{compactenum}[1)]
\item
$g(1)=b$,
\item
her $k$ sayma say\i s\i\ i\c cin $g(k+1)=f(g(k))$.
\end{compactenum}
\end{theorem}

\section{Ordinal say\i lar}

K\"ume aksiyomlar\i n\i\ kullanarak
ger\c cel say\i lar gibi 
\emph{ordinal say\i lar\i} in\c sa edebilece\u giz.
\c Simdilik onlar\i n var oldu\u gunu varsay\i yoruz.
\textbf{Ordinal say\i lar} veya \textbf{ordinaller,}
\begin{equation*}
\on
\end{equation*}
\emph{s\i n\i f\i n\i} olu\c sturur.
Her s\i n\i f,
$\{x\colon\phi(x)\}$
bi\c cimindedir, yani bir s\i n\i f\i n elemanlar\i,
\emph{tek serbest de\u gi\c skeni olan bir form\"ul\"u} sa\u glayan k\"umelerdir.
\"Ozel olarak her $a$ k\"umesi,
$\{x\colon x\in a\}$
s\i n\i f\i d\i r.

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

$\on$'nin aksiyomlar\i na g\"ore
\begin{compactenum}[1)]
\item
en az bir ordinal vard\i r;
\item
$\on$ iyis\i ral\i d\i r;
\item
her ordinal i\c cin, daha b\"uy\"uk ordinal vard\i r;
\item
$\on$'nin herhangi altk\"umesinin \"usts\i n\i r\i\ vard\i r;
\item
Herhangi $\alpha$ ordinali i\c cin $\{\xi\in\on\colon\xi<\alpha\}$ s\i n\i f\i\ bir k\"umedir.
\end{compactenum}
Burada $\alpha$, $\beta$, ve $\gamma$ sabitleri ve $\xi$ de\u gi\c skeni her zaman ordinal olacakt\i r.
\"Orne\u gin $\{\xi\in\on\colon\xi<\alpha\}=\{\xi\colon\xi<\alpha\}$.

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

\begin{proof}
Her ordinalin daha b\"uy\"u\u g\"u oldu\u gundan
$\on$'nin \"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{proof}

En k\"u\c c\"uk ordinal $0$
olarak kabul edilir.
Ayr\i ca, sayma say\i lar ordinal olarak kabul edilir,
ama $\R$'de ba\c ska ordinal yoktur.
Yani ger\c cel olan ordinaller, \emph{do\u gal say\i lard\i r.}
Herhangi $\alpha$ ordinali i\c cin, tan\i ma g\"ore
\begin{equation*}
\alpha'=\min\{\xi\colon\alpha<\xi\}.
\end{equation*}
Burada $\alpha'$, $\alpha$'n\i n \textbf{ard\i l\i d\i r.}
O zaman $\alpha$'n\i n ard\i l\i, 
$\alpha$'dan b\"uy\"uk olan ordinallerin en k\"u\c c\"u\u g\"ud\"ur.
\"Ornegin
\begin{align*}
	0'&=1,&
	1'&=2,&
	2'&=3,&
	3'&=4,
\end{align*}
ve saire.
Ne s\i f\i r ne bir ard\i l olan ordinal, bir \textbf{limittir.}

\begin{theorem}
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}

En k\"u\c c\"uk limit
\begin{equation*}
\upomega
\end{equation*}
olsun.
O zaman $\{\xi\colon\xi<\omega\}$, do\u gal say\i lar k\"umesidir.
S\i n\i flar, siyah harfler ile yazaca\u g\i z.

\begin{theorem}[Ordinal T\"umevar\i m]
$\bm A\included\on$ olsun.
E\u ger
\begin{compactenum}[1)]
\item
$0\in\bm A$,
\item
$\bm A$'n\i n her $\beta$ eleman\i\ i\c cin $\beta'\in\bm A$, ve
\item
her $\gamma$ limiti i\c cin
\begin{equation*}
\{\xi\colon\xi<\gamma\}\included\bm A\lto\gamma\in\bm A
\end{equation*}
\end{compactenum}
ise, o zaman $\bm A=\on$.
\end{theorem}

\begin{proof}
$\on\setminus\bm A$ fark\i n\i n en k\"u\c c\"uk eleman\i\ olamaz.
\end{proof}

Herhangi $\bm A$ s\i n\i f\i\ i\c cin
\begin{equation*}
\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.

\begin{theorem}[Ordinal \"Ozyineleme]
Bir $\bm A$ s\i n\i f\i\ i\c cin
\begin{compactenum}[1)]
\item
$b\in\bm A$, 
\item
$\bm F\colon\bm A\to\bm A$,
\item
ve $\bm G\colon\pow{\bm A}\to\bm A$ 
\end{compactenum}
olsun.
O zaman $\on$'den $\bm A$'ya giden bir ve tek bir $\bm H$ g\"ondermesi i\c cin
\begin{compactenum}[1)]
\item
$\bm H(0)=b$,
\item
her $\alpha$ ordinali i\c cin $\bm H(\alpha')=\bm F(\bm H(\alpha))$,
\item
her $\alpha$ limiti i\c cin
$\bm H(\alpha)=\bm G(\{\bm H(\xi)\colon\xi<\alpha\})$.
\end{compactenum}
\end{theorem}

\c Simdi $\bm F\colon\on\to\on$ olsun.  E\u ger
\begin{compactenum}[1)]
\item
$\bm F$ \textbf{kesin artan,} yani
$\alpha<\beta\lto\bm F(\alpha)<\bm F(\beta)$, 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{theorem}
$\bm F\colon\on\to\on$ ve kesin artan olsun.
O zaman $\bm F$ normaldir ancak ve ancak s\"ureklidir.
\end{theorem}

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

\subsection{Toplama}

Tan\i ma g\"ore her $\alpha$ ordinali i\c cin
\begin{gather*}
  \alpha+0=\alpha,\\
\alpha+\beta'=(\alpha+\beta)',\\
\gamma\text{ limit ise }\alpha+\gamma=\sup\{\alpha+\xi\colon\xi<\gamma\}.
\end{gather*}
\"Ozel olarak
\begin{equation*}
\alpha+1=\alpha'.
\end{equation*}

\begin{theorem}
Her $\alpha$ ordinali i\c cin $\xi\mapsto\alpha+\xi$ normaldir.
\end{theorem}

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

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

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

\begin{theorem}\label{thm:+assoc}
Ordinaller toplamas\i\ birle\c smelidir.
\end{theorem}

Ordinal toplama de\u gi\c smeli de\u gildir \c c\"unk\"u
\begin{equation*}
1+\upomega=\sup_{x<\upomega}(1+x)=\upomega<\upomega+1.
\end{equation*}

\subsection{\c Carpma}

Tan\i m\i na 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*}
\"Ozel olarak
\begin{equation*}
\alpha\cdot1=\alpha.
\end{equation*}
O zaman $\upomega\cdot2 
=\upomega\cdot1+\upomega
=\upomega+\upomega
=\sup\{\upomega+x\colon x\in\upomega\}$, ama
\begin{equation*}
2\cdot\upomega=\sup_{x\in\upomega}(2\cdot x)=\upomega,  
\end{equation*}
dolay\i s\i yla $2\cdot\upomega<\upomega\cdot2$.
\"Oyleyse \c carpma de\u gi\c smeli de\u gildir.

\begin{theorem}
$0\cdot\alpha=0$ ve $1\cdot\alpha=\alpha$.
\end{theorem}

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

\begin{theorem}
Ordinaller \c carpmas\i, toplama \"uzerine soldan da\u g\i l\i r,
yani
\begin{equation*}
 \alpha\cdot(\beta+\gamma)=\alpha\cdot\beta+\alpha\cdot\gamma. 
\end{equation*}
\end{theorem}

\begin{theorem}
Ordinaller \c carpmas\i\ birle\c smelidir.
\end{theorem}

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

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

\subsection{Kuvvet alma}

Her $\alpha$ i\c cin, $\alpha>0$ ise, tan\i m\i na g\"ore,
\begin{gather*}
	\alpha^0=1,\\
	\alpha^{\beta'}=\alpha^\beta\cdot\alpha,\\
	\gamma\text{ limit ise }\alpha^\gamma=\sup\{\alpha^{\xi}\colon\xi<\gamma\}.
\end{gather*}%
\"Ozel olarak
\begin{equation*}
\alpha^1=\alpha.
\end{equation*}
Ayr\i ca, tan\i ma g\"ore,
\begin{align*}
0^0&=1,&\beta>0&\lto 0^{\beta}=0.
\end{align*}

\begin{theorem}\mbox{}\label{thm:powers}
  \begin{compactenum}
  \item 
$1^{\alpha}=1$.
\item
$\alpha\geq1$ ise $\xi\mapsto\xi^{\alpha}$ artand\i r.
\item
$\alpha\geq2$ ise $\xi\mapsto\alpha^{\xi}$ i\c slemi, normaldir.
\item
$\alpha^{\beta+\gamma}=\alpha^{\beta}\cdot\alpha^{\gamma}$.
\item
$\alpha^{\beta\cdot\gamma}=(\alpha^{\beta})^{\gamma}$.
  \end{compactenum}
\end{theorem}


\end{document}