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\begin{document}
%\frontmatter
\title{Ordinal Analiz}
\subtitle{Aksiyomatik K\"umeler Kuram\i\ Dersi}
\author{David Pierce}
\date{22 \c Subat 2018}
%\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

%\frontmatter

%\selectlanguage{english}

\mainmatter

%\selectlanguage{turkish}

\tableofcontents

\listoffigures

%\mainmatter

\chapter{Ger\c cel Analiz}\label{ch:real}

\section{Tam s\i ral\i\ cisim aksiyomlar\i}

Ger\c cel say\i lar,
$\R$\label{R} \textbf{tam s\i ral\i\ cismini} olu\c sturur.
Demek ki
\begin{compactenum}[1)]
\item
$\R$, $<$ ba\u g\i nt\i s\i\ taraf\i ndan \textbf{s\i ralanm\i\c st\i r,} yani
\begin{compactenum}
\item
$<$ ba\u g\i nt\i s\i\ \textbf{yans\i mas\i zd\i r,}
\begin{equation*}
a\not< a;
\end{equation*}
\item
$<$ ba\u g\i nt\i s\i\ \textbf{ge\c ci\c slidir,}
\begin{equation*}
a<b\land b<c\lto a<c;
\end{equation*}
\end{compactenum}
\item
$<$ s\i ralamas\i\ \textbf{do\u grusald\i r,}
\begin{equation*}
a<b\lor a=b\lor a>b;
\end{equation*}
\item
$<$ do\u grusal s\i ralamas\i\ \textbf{tamd\i r,}
yani $\R$'nin bo\c s olmayan, \"usts\i n\i r\i\ olan her altk\"umesinin
en k\"u\c c\"uk \"usts\i n\i r\i\
veya \textbf{supremumu} vard\i r:
\begin{multline*}\label{A}
\Exists xx\in A\land\Exists x\Forall y(y\in A\lto y\leq x)\lto\\
\Exists x\Bigl(\Forall y(y\in A\lto y\leq x)\land{}\\
\Forall z\bigl(\Forall y(y\in A\lto y\leq z)\lto x\leq z\bigr)\Bigr);	
\end{multline*}
\item
$\R$, iki-konumlu \textbf{toplama} ve \textbf{\c carpma} i\c slemleri alt\i nda kapal\i d\i r,
ve bu i\c slemler ile $\R$ bir \textbf{cisimdir,} yani
\begin{gather*}
\begin{gathered}
	a+b=b+a,\\
	a+0=a,\\
	-a+a=0,	
\end{gathered}\qquad
\begin{gathered}
	ab=ba,\\
	a\cdot 1=a,\\
	a\neq0\lto\Exists xax=1,
	\end{gathered}\\
	a\cdot(b+c)=ab+ac;
\end{gather*}
\item
$\R$'nin s\i ralamas\i\ ve cisim yap\i s\i\ birbirine sayg\i\ g\"osterir:
\begin{gather*}
	a\leq0\liff-a\geq0,\\
	a>0\land b>0\lto a+b>0\land ab>0.
\end{gather*}
\end{compactenum}
Her iki tam s\i ral\i\ cismin birbirine izomorf oldu\u gunu,
teorem olarak
kan\i tlayabiliriz.

\section{Ger\c cel say\i lar\i n in\c sas\i}

Ger\c cel analizdeki gibi,
bu b\"ol\"umde $\R$'nin 
%tam s\i ral\i\ cisminin 
var oldu\u gunu,
aksiyom olarak kabul ediyoruz.
Fakat \emph{k\"ume aksiyomlar\i n\i} kullanarak
ger\c cel say\i lar\i\ in\c sa edebiliriz.
K\i saca
\begin{compactenum}[1)]
\item\label{Q} 
$\R$,
$\Q$ kesirli say\i lar s\i ral\i\ cisminden elde edilir,
\item\label{Z}
$\Q$, $\Z$ tamsay\i lar s\i ral\i\ halkas\i ndan elde edilir,
\item\label{N}
$\Z$, $\N$ sayma say\i lar\i\ yap\i s\i ndan elde edilir.
\end{compactenum}
Yukar\i daki in\c salar,
a\c sa\u g\i daki \c sekilde yap\i l\i r.
\begin{compactenum}
\item
Her ger\c cel say\i, \"oyle bir $A$ k\"umesi olur ki
\begin{compactenum}
\item
$\emptyset\pincluded A\pincluded\Q$,
yani $A$ bo\c s de\u gildir,
$A$'n\i n elemanlar\i\ kesirli say\i d\i rlar,
ve her kesirli say\i\ $A$'n\i n eleman\i\ de\u gildir;
\item
$\Forall x\Forall y(x\in A\land y\in\Q\land y<x\lto y\in A)$,
yani $A$'n\i n bir eleman\i ndan k\"u\c c\"uk olan her kesirli say\i\
$A$'n\i n eleman\i d\i r; ve
\item
$\Forall x\Exists y(x\in A\lto y\in A\land x<y)$,
yani $A$'n\i n en b\"uy\"uk eleman\i\ yoktur.
\end{compactenum}
\item
Her $a/b$ kesirli say\i s\i,
$\{(x,y)\in\Z\times(\Z\setminus\{0\})\colon ay=bx\}$
k\"umesi olarak tan\i mlan\i r.
\item
Benzer \c sekilde her tamsay\i,
baz\i\ $a$ ve $b$ sayma say\i lar\i\ i\c cin
$a-b$ bi\c ciminde yaz\i labilir,
ve tamsay\i n\i n kendisi
$\{(x,y)\in\N\times\N\colon a+y=b+x\}$
k\"umesi olarak tan\i mlan\i r.
\end{compactenum}
Burada $\N$ yap\i s\i n\i n \"ozelliklerini,
\emph{Peano Aksiyomlar\i ndan} t\"uretebiliriz,
ve ondan sonra $\R$'nin tam s\i ral\i\ bir cisim oldu\u gunu,
teorem olarak kan\i tlayabiliriz.

Peano Aksiyomlar\i n\i\ kullanmadan
$\N$, 
s\i f\i r olmayan sonlu olan \emph{ordinal say\i lar} taraf\i ndan olu\c sturulabilir.
B\"ol\"um \numarada{ch:ord} ordinallerin \"ozelliklerini,
aksiyom olarak verece\u giz.
B\"ol\"um \numarada{ch:ax}
her ordinali bir k\"ume olarak tan\i mlayaca\u g\i z,
ve ordinallerin
\emph{ordinal aksiyomlar\i} sa\u glad\i\u g\i n\i\,
teorem olarak kan\i tlayaca\u g\i z.
Bu \c sekilde ger\c cel analizi,
k\"umeler kuram\i nda temelle\c stirebiliriz.
Ayr\i ca ger\c el analizin
ve ordinal analizin baz\i\ ortak \"ozellikleri
olacakt\i r.

\section{Sayma say\i lar\i}

\c Simdilik, tam tersine,
ger\c cel say\i lar\i n yukar\i daki aksiyomlar\i n\i\ varsayarak
$\N$ yap\i s\i n\i\ elde edece\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 ma g\"ore
\begin{equation*}
\N=\bigcap\{X\included\R\colon X\text{ t\"umevar\i ml\i d\i r}\}
\end{equation*}
olsun.
Bu \c sekilde \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.
Genelde elemanlar\i\ k\"ume olan her $\mathscr B$ k\"umesi i\c cin
\begin{equation*}
\bigcap\mathscr B=\{x\colon\Forall Y(Y\in\mathscr B\lto x\in Y)\}.
\end{equation*}
Bu yeni k\"ume, $B$'nin \textbf{kesi\c simidir.}
\"Ozel olarak
\begin{equation*}
C\cap D=\bigcap\{C,D\}.
\end{equation*}
Ayr\i ca
\begin{equation*}
\bigcap_{i\in\N}A_i=\bigcap\{A_i\colon i\in\N\}.
\end{equation*}

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

\ktk

Bu teoreme g\"ore t\"umevar\i ml\i\ kan\i tlar yap\i labilir.
Yani $\N$'nin herhangi $A$ altk\"umesi i\c cin
\begin{compactenum}[1)]
\item
$1\in A$ ve
\item
$b\in A\lto b+1\in A$
%$\Forall x(x\in A\lto x+1\in A)$
\end{compactenum}
ise, o zaman t\"umevar\i mdan $A=\N$.
Bu kan\i tta $b\in A$ varsay\i m\i,
kan\i t\i n \textbf{t\"umevar\i m hipotezidir.}

\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}

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

Her $a$ ger\c cel say\i s\i, $(a-1)+1$ bi\c ciminde yaz\i labilir,
ama $a$'n\i n sayma say\i s\i\ oldu\u gunda bile $a-1$, sayma say\i s\i\ olmayabilir.

\begin{lemma}\label{lem:N<}
	$\N$ do\u grusal s\i ral\i d\i r, ve her $k$ eleman\i\ i\c cin
	\begin{equation*}
	k<k+1.
	\end{equation*}
	Ayr\i ca
	\begin{equation*}
	k<\ell\lto k+1<\ell+1.
	\end{equation*}
\end{lemma}

\klk[ ($\N\included\R$ oldu\u gundan $\N$,
$\R$'den baz\i\ \"ozellikleri al\i r.)]

\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}

\klk[ (T\"umevar\i m ve  Lemma \numarayi{lem:N<}
kullan\i n.)]

\begin{lemma}\label{lem:k<m}
Herhangi $k$ ve $m$ sayma say\i lar\i\ i\c cin
\begin{equation}\label{eqn:km}
k\leq m\liff k<m+1,
\end{equation}
yani
$\{x\in\N\colon x<m\}\cup\{m\}=\{x\in\N\colon x<m+1\}$.
\end{lemma}

\klk[ (Lemma \numarada{lem:N<}n
\eqref{eqn:km} denkli\u gi
$m<k\liff m+1\leq k$ bi\c ciminde yaz\i labilir.
Bunun bir y\"on\"u apa\c c\i kt\i r.
Di\u ger y\"on $k$ \"uzerinde t\"umevar\i m,
Lemmalar \ref{lem:-}, \ref{lem:N<}, ve \ref{lem:1leq} ile
kan\i tlanabilir.)]

\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}

\ktk[ ($\bigl\{x\in\N\colon\{y\in\N\colon y<x\}\included A\bigr\}$ k\"umesi $B$ olsun.
Lemmalar \ref{lem:1leq} ve \ref{lem:k<m}
ve t\"umevar\i m ile $B=\N$ oldu\u gunu kan\i tlay\i n.)]

\"Orne\u gin 
g\"u\c cl\"u t\"umevar\i mdan
her sayma say\i s\i\ ya $1$'dir
ya da asal bir say\i\ taraf\i ndan b\"ol\"un\"ur.
Zira bu \"ozelli\u gi olan sayma say\i lar\i\
bir $A$ k\"umesini olu\c stursun.
Bir $m$ i\c cin $k<m$ ise $k\in A$ olsun.
E\u ger $m=1$ ise $m\in A$.
E\u ger $m$ asal ise $m\in A$.
Kalan durumda bir $k$ i\c cin $1<k<m$ ve $k\divides m$.
(Burada
O zaman $k\in A$, ama $k\neq1$,
dolay\i s\i yla bir $p$ asal\i\ i\c cin $p\divides k$,\label{p}
ve sonu\c c olarak $p\divides m$ ve $m\in A$.
G\"u\c cl\"u t\"umevar\i mdan $A=\N$.

\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}

\ktk[ ($A\included\N$ olsun,
ama $A$'n\i n en k\"u\c c\"uk eleman\i\ olmas\i n.
G\"u\c cl\"u t\"umevar\i m ile $\N\setminus A=\N$ kan\i tlay\i n.)]

\section{G\"ondermeler}

E\u ger $f$, tan\i m k\"umesi $A$ olan
ve de\u ger k\"umesi $B$ olan bir g\"ondermeyse,
bu durum
\begin{equation*}
  f\colon A\to B
\end{equation*}
c\"umlesiyle ifade edilebilir.
Ayr\i ca $f$'nin kendisinin yerine
\begin{equation*}
x\mapsto f(x)
\end{equation*}
isimi kullan\i labilir;
\Sekle{fig:rec}, Al\i\c st\i rma \numaraya{ex:xx'} ve \Teoreme{thm:a+x} bak\i n.

E\u ger $f\colon A\to B$, $C\included A$, $g\colon C\to B$, ve $C$'nin her $d$ eleman\i\ $g(d)=f(d)$ ise,
o zaman
\begin{equation*}
g=f\restriction C;
\end{equation*}
\Teoremin{thm:ord-rec} kan\i t\i na bak\i n.
E\u ger $A$'dan $B$'ye giden birebir ve \"orten g\"onderme varsa,
bu g\"onderme bir \textbf{e\c slemedir,}
ve verilen k\"umeler \textbf{e\c sleniktir;}
bu durum
\begin{equation*}
  A\approx B
\end{equation*}
c\"umlesiye ifade edilir.
\Teoreme{thm:a+b} bak\i n.

Herhangi $A$ ve $B$ k\"umelerinin \textbf{kartezyan \c carp\i m\i} vard\i r.
Tan\i ma g\"ore
\begin{equation*}
A\times B=\{(x,y)\colon x\in A\land y\in B\}.
\end{equation*}
Burada
\begin{equation*}
(a,b)=(c,d)\liff a=c\land b=d.
\end{equation*}
E\u ger $f\colon A\to B$ ise,
o zaman $f$, $A\times B$ \c carp\i m\i n\i n
\begin{equation*}
\bigl\{\bigl(x,f(x)\bigr)\colon x\in A\bigr\}
\end{equation*}
altk\"umesini belirtir.

Bir $A$ k\"umesinde tek-konumlu bir i\c slem,
$A$'dan kendisine giden bir g\"ondermedir;
iki-konumlu bir i\c slem,
$A\times A$ \c carp\i m\i ndan $A$'ya giden bir g\"ondermedir.

Ger\c cel analiz ve say\i lar kuram\i nda
tan\i m k\"umesi $\N$ olan g\"ondermeler tan\i mlan\i p kullan\i l\i r.
\"Orne\u gin $x\mapsto x!$ g\"ondermesi i\c cin
\begin{align*}
	1!&=1,&(k+1)!&=(k+1)\cdot k!
\end{align*}
\textbf{\"ozyineli} tan\i m\i\ verilir.
Bu tan\i m neden ge\c cerli midir?

Tan\i m\i n ge\c cerlili\u gi i\c cin
$\N$ t\"umevar\i ml\i\ olmal\i d\i r,
ama bunu \Teoremde{thm:ind}n biliyoruz.
Ayr\i ca $\N$, 
ger\c cel say\i lar\i n \c carpmas\i\ alt\i nda kapal\i\ olmal\i d\i r.

\begin{theorem}
T\"um $a$ ve $b$ ger\c cel say\i lar\i\ i\c cin
\begin{equation*}
a\in\N\land b\in\N\lto a+b\in\N\land a\cdot b\in\N.
\end{equation*}
\end{theorem}

\ktk

\c Simdi $1\in\N$,
ve ayr\i ca $\N$'nin herhangi $k$ eleman\i\ i\c cin
e\u ger $k!\in\N$ ise,
o zaman $(k+1)\cdot k!\in\N$.
Bu \c sekilde $x\mapsto x!$ g\"ondermesi
tan\i mlanabilir mi?
\begin{compactitem}
\item
T\"umevar\i m veya g\"u\c cl\"u t\"umevar\i m ile 
bir k\"umenin $\N$ oldu\u gu kan\i tlanabilir;
ama $\{x\in\N\colon x!\text{ tan\i mlan\i r}\}$,
iyi tan\i mlanm\i\c s bir k\"ume de\u gildir.
\item
\.Iyis\i ralama ile $\N$'nin bo\c s olmayan bir alt\"umesinin eleman\i\ bulunabilir;
ama $x\mapsto x!$ g\"ondermesi, $\N$'nin bir eleman\i\ de\u gildir.
\end{compactitem}
Ba\c ska bir teoreme ihtiyac\i m\i z vard\i r.


\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 $\N$'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}
\Sekle{fig:rec} bak\i n.
\begin{figure}
\relscale{1.1}
\begin{equation*}
\xymatrix@=2cm{\{1\}\ar[d]_g\ar[r]^{\included}&\N\ar[d]_g\ar[r]^{x\mapsto x+1}&\N\ar[d]^g\\
\{b\}\ar[r]_{\included}&A\ar[r]_f&A}
\end{equation*}
\caption{\"Ozyineleme}\label{fig:rec}
\end{figure}
\end{theorem}

\"Orne\u gin $A=\N\times\N$,
$b=(1,1)$, ve
\begin{equation*}
f(x,y)=\bigl(x+1,(x+1)\cdot y\bigr)
\end{equation*}
olsun.
O zaman bir ve tek bir $g$ g\"ondermesi i\c cin
$g$'nin tan\i m k\"umesi $\N$, $g(1)=(1,1)$, ve $g(k+1)=f(g(k))$.
\c Simdi $g(k)$, $(g_1(k),g_2(k))$ olarak yaz\i ls\i n.
T\"umevar\i mdan $g_1(k)=k$.  Bundan dolay\i
\begin{equation*}
g_2(k+1)=(k+1)\cdot g_2(k).
\end{equation*}
Ayr\i ca $g_2(1)=1$.
B\"oylece $k!$, $g_2(k)$ olarak tan\i mlanabilir.

\begin{proof}[\"Ozyineleme Teoremi kan\i t\i.]
Bir ve tek bir $h$ g\"ondermesi i\c cin
\begin{compactenum}[1)]
\item
g\"ondermenin tan\i m k\"umesi, $\N$'nin tek-elemanl\i\ $\{1\}$ altk\"umesidir, ve
\item
$h(1)=b$.
\end{compactenum}
Bu g\"onderme $h_1$ olsun.

T\"umevar\i m hipotezi olarak
bir $m$ sayma say\i s\i\ i\c cin,
bir ve tek bir $h$ g\"ondermesi i\c cin
\begin{compactenum}[1)]
\item
g\"ondermenin tan\i m k\"umesi, $\N$'nin $m$ elemanl\i\ $\{1,\dots,m\}$ altk\"umesi olsun,
\item
$h(1)=b$ olsun, ve
\item
$k<m$ ise $h(k+1)=f(h(k))$ olsun.
\end{compactenum}
Bu g\"onderme $h_m$ olsun.
O zaman
bir ve tek bir $h$ g\"ondermesi i\c cin
\begin{compactenum}[1)]
\item
g\"ondermenin tan\i m k\"umesi, $\{1,\dots,m+1\}$ altk\"umesidir,
\item
$h(1)=b$, ve
\item
$k<m+1$ ise $h(k+1)=f(h(k))$.
\end{compactenum}
Zira b\"oyle bir $h$ varsa,
o zaman $h\restriction\{1,\dots,m\}$ ve $h_m$ g\"ondermelerinin
\"ozellikleri ayn\i d\i r,
dolay\i s\i yla $h_m$ g\"ondermesinin biricikli\u ginden $h\restriction\{1,\dots,m\}=h_m$.
Bu \c sekilde $h$'nin tan\i m\i
\begin{equation}\label{eqn:h}
h(x)=\begin{cases}
	h_m(x),&\text{ $x\leq m$ durumunda,}\\
	f(h_m(m)),&\text{ $x=m+1$ durumunda}
\end{cases}
\end{equation}
olabilir.
Ayr\i ca $h\restriction\{1,\dots,m\}=h_m$ olmal\i d\i r,
dolay\i s\i yla $h$'nin kendisi,
\eqref{eqn:h} e\c sitli\u gini sa\u glamal\i d\i r.
Bu $h$ g\"ondermesi $h_{m+1}$ olsun.

T\"umevar\i mdan,
her $n$ sayma say\i s\i\ i\c cin,
$\{1,\dots,n\}$ k\"umesinden giden bir be tek bir $h_n$ g\"ondermesi i\c cin
$h_n(1)=b$ ve $k<n$ ise $h_n(k+1)=f(h_n(k))$.
Ayr\i ca
\begin{equation*}
h_{m+1}(m+1)=f(h_m(m)).
\end{equation*}
\c Simdi $g(x)=h_x(x)$ olsun.  O zaman
\begin{gather*}
g(1)=h_1(1)=b,\\
g(k+1)=h_{k+1}(k+1)=f(h_k(k))=f(g(k)).
\end{gather*}
Bu \c sekilde $g$, istedi\u gimiz gibidir.
Bir $h$ g\"ondermesinin istedi\u gimiz \"ozelli\u gi varsa
\begin{equation*}
h(1)=b=g(1),
\end{equation*}
ve $h(m)=g(m)$ ise
\begin{equation*}
	h(m+1)=f(h(m))=f(g(m))=g(m+1).
\end{equation*}
Bu \c sekilde her $k$ sayma say\i s\i\ i\c cin $h(k)=g(k)$,
dolay\i s\i yla $h=g$.	
\end{proof}

\begin{sloppypar}
  Baz\i\ yap\i larda t\"umevar\i m kullan\i labilir,
  ama \"ozyineleme kullan\i lamaz.
  \"Orne\u gin $p$ asal ise,\label{p2}
  Fermat Teoremine g\"ore herhangi $a$ tamsay\i s\i\ i\c cin
  \begin{equation}\label{eqn:Fermat}
    a^p\equiv a\pmod p.
  \end{equation}
  T\"umevar\i m ile bu teoremi kan\i tlayabiliriz,
  zira $1^p\equiv1$, 
  ve ayr\i ca $b^p\equiv b$ ise, o zaman
  \begin{align*}
    (b+1)^p
    &=b^p+pb^{p-1}+\binom p2b^{p-2}+\dots+\binom p{p-2}b^2+pb+1\\
    &\equiv b^p+1\equiv b+1\pmod p,
  \end{align*}
  \c c\"unk\"u $0<k<p$ ise $k\divides\binom pk$.
  Neden bu kan\i t ge\c cerlidir?
  Say\i lar kuram\i ndan
  \begin{multline}\label{eqn:+.mod}
    a\equiv a_1\land b\equiv b_1\\
    \lto a+b\equiv a_1+b_1\land ab\equiv a_1b_1\pmod p.
  \end{multline}
  $\Zmod p$, tamsay\i lar\i n $p$'ye g\"ore kalanda\c sl\i k s\i n\i flar\i\ k\"umesi olsun.
  Bu \c sekilde
  \begin{align*}
    \Zmod p&=\{[x]\colon x\in\Z\},&
	  [k]&=\{x\in\Z\colon x\equiv k\pmod p\}.
  \end{align*}
  O zaman $\Zmod p=\{[1],\dots,[p]\}$.
  Ayr\i ca \eqref{eqn:+.mod} gerektirmesine g\"ore
  \begin{align*}
    [a]+[b]&=[a+b],&
    [a][b]&=[ab]
  \end{align*}
  tan\i mlar\i\ ge\c cerlidir, \c c\"unk\"u
  \begin{equation*}
    [a]=[a_1]\land[b]=[b_1]\lto[a+b]=[a_1+b_1]\land[ab]=[a_1b_1].
  \end{equation*}
  \c Simdi $A\included\Zmod p$ ve $[1]\in A$ olsun,
  ve $[k]\in A$ ise $[k+1]\in A$ olsun.
  O zaman t\"umevar\i mdan $A=\Zmod p$.
  Zira $B=\{x\in\N\colon[x]\in A\}$ olsun.
  O zaman $1\in B$, \c c\"unk\"u $[1]\in A$.
  Ayr\i ca $k\in B$ ise,
  o zaman $[k]\in A$, dolay\i s\i yla $[k+1]\in A$ ve $k+1\in B$.
  T\"umevar\i mdan $B=\N$.
  \"Ozel olarak $\{1,\dots,p\}\included B$,
  dolay\i s\i yla $\Zmod p=A$.
\end{sloppypar}
Yukar\i daki g\"osterdi\u gimize g\"ore $[1]^p=[1]$,
ve $[b]^p=[b]$ ise $[b+1]^p=[b+1]$.
O zaman t\"umevar\i mdan her $a$ tamsay\i s\i\ i\c cin $[a]^p=[a]$,
yani \eqref{eqn:Fermat} kalanda\c sl\i\u g\i\ do\u grudur.

B\"oylece $\Zmod p$ yap\i s\i nda t\"umevar\i m y\"ontemi ge\c cerlidir;
ama \"ozyineleme y\"ontemi ge\c cerli de\u gildir.
\"Orne\u gin $\Zmod3$ yap\i s\i nda hi\c cbir tek-konumlu $g$ i\c slemi i\c cin
\begin{align*}
	g([1])&=[2],&g([k+1])&=[k][2]
\end{align*}
olmaz, \c c\"unk\"u olursa
\begin{align*}
	g([2])&=[4]=[1],&
	g([3])&=[2],&
	g([4])&=[1],
\end{align*}
ama $[4]=[1]$ oldu\u gundan $g([4])=g([1])=[2]$,
ve $[2]\neq[1]$.

\begin{xca}\sloppy
\"Ozyineleme Teoreminin yukar\i daki kan\i t\i,
$\N$'nin hangi \"ozelliklerini kullan\i r?
\end{xca}

\section{Peano Aksiyomlar\i}

\"Ozyineleme Teoreminin ba\c ska bir kan\i t\i\ vard\i r.

\begin{proof}[\"Ozyineleme Teoremi ikinci kan\i t\i.]
Birinci kan\i ttaki gibi,
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 
$\N$'nin bir altk\"umesidir, ve
\item
herhangi $\ell$ sayma say\i s\i\ i\c cin,
$h(\ell)$ tan\i mlan\i rsa, o zaman
\begin{compactenum}
\item 
ya $\ell=1$ ve $h(\ell)=b$,
\item
ya da bir $k$ sayma 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}
Lemma \ref{lem:-} sayesinde istedi\u gimiz gibi $g$ g\"ondermesi varsa
$\mathscr C$'nin eleman\i d\i r.
Her $k$ sayma 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 $k$ sayma say\i lar\i,
$E$ k\"umesini olu\c stursun.
Tan\i m k\"umesi $\{1\}$ olan
bir $h$ g\"ondermesi i\c cin $h(1)=b$.
O zaman $h\in\mathscr C$.
Ayr\i ca $\mathscr C$'nin herhangi $h$ eleman\i\ i\c cin $h(1)$ tan\i mlan\i rsa,
o zaman $h(1)=b$ olmal\i d\i r, 
\c c\"unk\"u hi\c c $k$ sayma say\i s\i\ i\c cin $k+1=1$ de\u gildir.
Bu \c sekilde $1\in E$.

\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 $k+1\neq1$, ve ayr\i ca herhangi $\ell$ sayma 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 $k+1\neq1$ oldu\u gundan bir $\ell$ sayma 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 $E=\N$.
\end{proof}

\begin{sloppypar}
  Yukar\i daki kan\i t, sadece $\N$'nin a\c sa\u g\i daki \"ozelliklerini kullan\i r:
  \begin{compactenum}
  \item
    $1\in\N$.
  \item
    $k\in\N$ ise $k+1\in\N$.
  \item
    T\"umevar\i m y\"ontemi ge\c cerlidir.
  \item
    Her $k$ sayma say\i s\i\ i\c cin
    $1\neq k+1$.
  \item
    T\"um $k$ ve $\ell$ sayma 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,
  $\N$'de 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 
  $\N$'de 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 $\N$'nin s\i ralamas\i n\i\ tan\i mlay\i p
  \"ozelliklerini kan\i tlayabiliriz.
  Ondan sonra
  yukar\i daki gibi $\Z$, $\Q$ ve $\R$ yap\i lar\i n\i\ elde edebiliriz.
\end{sloppypar}
Tam tersine tam s\i ral\i\ cisim aksiyomlar\i n\i\ kullanarak
$\N$ yap\i s\i n\i\ in\c sa ettik
ve onun
Peano Aksiyomlar\i n\i\ sa\u glad\i\u g\i n\i\ teorem olarak kan\i tlad\i k.
(\Teorem{thm:ind} ve Lemmalar \ref{lem:N<} ve \numaraya{lem:1leq} bak\i n.)

\begin{sloppypar}
  Sayma say\i lar\i na s\i f\i r\i\ ekleyerek do\u gal say\i lar\i\ elde ederiz.
  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}

K\"umeler kuram\i m\i zda her k\"umenin her eleman\i,
bir k\"ume olacakt\i r.
(Bu \c sekilde
``elemanlar\i\ k\"ume olan k\"ume'' ifadesi
gereksiz k\i l\i nacakt\i r.)
Bir k\"ume bo\c s olacakt\i r,
ve bu k\"ume,
\begin{equation*}
\emptyset
\end{equation*}
olarak yaz\i l\i r.
Ayr\i ca
\begin{equation}\label{eqn:0=emptyset}
0=\emptyset
\end{equation}
tan\i mlayaca\u g\i z.
Herhangi $a$ k\"umesi i\c cin
\begin{equation}\label{eqn:a'=}
a'=a\cup\{a\}
\end{equation}
tan\i mlayaca\u g\i z.
Ayr\i ca
\begin{align}\label{eqn:1=0'}
	1&=0',&2&=1',&3&=2',&4&=3',&&\dots
\end{align}
olacakt\i r.
O zaman $0$, $1$, $2$, $3$, \lips, 
\textbf{do\u gal say\i} olacaklard\i r.
Do\u gal say\i lar, sonlu \textbf{ordinal} olacaklard\i r,
ama sonsuz ordinaller de var olacakt\i r.
\"Orne\u gin
\begin{equation}\label{eqn:omega=}
\upomega=\{0,1,2,\dots\},
\end{equation}
ve $\upomega$, en k\"u\c c\"uk sonsuz ordinal olacakt\i r.

\c Simdilik \eqref{eqn:0=emptyset}, \eqref{eqn:a'=}, ve \eqref{eqn:omega=}
tan\i mlar\i n\i\ kullanmayaca\u g\i z.
B\"ol\"um \numarada{ch:ax},
k\"ume aksiyomlar\i n\i\ kullanarak,
ordinalleri tan\i mlay\i p
\"ozelliklerini
teorem olarak kan\i tlayaca\u g\i z;
ama
\c simdilik ordinaller\i n a\c sa\u g\i da verilen \"ozelliklerini
aksiyom olarak kabul edece\u giz.

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

Ordinaller bir
\begin{equation*}
\on
\end{equation*}
\textbf{s\i n\i f\i n\i} olu\c sturacakt\i r.
Biz zaten $\Zmod p$ yap\i s\i n\i\ tan\i mlamak i\c cin \emph{denklik s\i n\i flar\i} kulland\i k.
Normalde bir denklik s\i n\i f\i\ bir k\"umedir.
Asl\i nda her k\"ume bir s\i n\i ft\i r,
ama her s\i n\i f bir k\"ume de\u gildir.

Her s\i n\i f
\emph{tek serbest de\u gi\c skeni olan bir form\"ul} 
taraf\i ndan tan\i mlan\i r.
\"Orne\u gin birazdan kullanaca\u g\i m\i z
\begin{gather*}
	x\in a,\\
	x\in\bm A\liff x\in\bm B,\\
	x\in c\liff x\in d,\\
	x\notin x
\end{gather*}
ifadeleri,
serbest de\u gi\c skeni $x$ olan form\"uld\"urler.
(Form\"ullerin resmi tan\i m\i\ i\c cin,
\ref{mantik} Eki'ne bak\i n.)
E\u ger $\phi$,\label{phi} 
tek serbest de\u gi\c skeni olan bir form\"ul ise,
o zaman $\phi$'nin tan\i mlad\i\u g\i\ s\i n\i f\i n elemanlar\i,
$\phi$'yi sa\u glayan k\"umelerdir,
ve s\i n\i f\i n kendisi
\begin{equation*}
\{x\colon\phi(x)\}
\end{equation*}
olarak yaz\i labilir.
S\i n\i flar b\"uy\"uk siyah harfler de ile g\"osterece\u giz.
K\"u\c c\"uk harfler
% ve siyah olmayan b\"uy\"uk harfler,
her zaman k\"ume olacakt\i r.
\"Ornegin
\begin{equation*}
\bm A=\{x\colon\phi(x)\}
\end{equation*}
ise, o zaman her $b$ k\"umesi i\c cin
\begin{equation*}
b\in\bm A\liff\phi(b).
\end{equation*}
\emph{Her k\"ume bir s\i n\i fa e\c sittir.}
\"Ozel olarak her $a$ k\"umesi i\c cin
\begin{equation*}
a=\{x\colon x\in a\}.
\end{equation*}
Genelde elemanlar\i\ ayn\i\ olan s\i n\i flar ve k\"umeler e\c sittir:
\begin{gather*}
\bm A=\bm B\liff\Forall x(x\in\bm A\liff x\in\bm B),\\
c=d\liff\Forall x(x\in c\liff x\in d).	
\end{gather*}

\begin{theorem}[Russell Paradoksu]
Her s\i n\i f bir k\"umeye e\c sit de\u gildir.
\"Orne\u gin $\{x\colon x\notin x\}$ s\i n\i f\i\ bir k\"umeye e\c sit de\u gildir.
\end{theorem}

\begin{proof}
$x\notin x$ form\"ul\"u $\phi(x)$ olarak yaz\i ls\i n.  E\u ger $\{x\colon\phi(x)\}=a$ ise,
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}

\"Oklid'de e\c sitlik, ayn\i l\i k de\u gildir.
\.Ikizkenar bir \"u\c cgenin iki e\c sit kenar\i\ vard\i r.
Bu kenarlar iki oldu\u gundan birbiriyle ayn\i\ de\u gildir.
Ama e\c sit s\i n\i flar ayn\i\ olarak d\"u\c s\"un\"ulebilir.

\section{Ordinallerin \"ozellikleri}

K\"u\c c\"uk Yunan harfleri her zaman ordinal g\"osterecektir.
\"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*}
\c Simdilik aksiyom olarak kabul edece\u gimiz
$\on$'nin \"ozellikleri a\c sa\u g\i dad\i r.
\begin{enumerate}
\item\label{on-props}
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 $\on$'de olan \"usts\i n\i r\i\ vard\i r.
\item
Herhangi $\alpha$ i\c cin $\{\xi\colon\xi<\alpha\}$ s\i n\i f\i\ bir k\"umedir.
\item
Herhangi $\bm F$ tek-konumlu ordinal i\c slemi i\c cin,
herhangi $\alpha$ i\c cin $\{\bm F(\xi)\colon\xi<\alpha\}$ s\i n\i f\i\ bir k\"umedir.
\item
Bir $\alpha$ i\c cin $\{\xi\colon\xi<\alpha\}$ k\"umesi sonsuzdur.
\end{enumerate}
Asl\i nda her $\alpha$ ordinali,
$\{\xi\colon\xi<\alpha\}$ k\"umesinin kendisi olarak tan\i mlanabilecektir;
ama \c simdilik bu tan\i m\i\ kullanmay\i p
sadece yukar\i daki yedi \"ozelli\u gi kullanaca\u g\i z.

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

\begin{proof}
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{proof}

\c Simdi $0$,
en k\"u\c c\"uk ordinal olarak tan\i mlans\i n,
ve herhangi $\alpha$ ordinali i\c cin
\begin{equation}\label{eqn:a'=min}
\alpha'=\min\{\xi\colon\alpha<\xi\}
\end{equation}
tan\i mlans\i n.
Bu \c sekilde $\alpha'$, 
$\alpha$'n\i n \textbf{ard\i l\i,} yani
$\alpha$'dan b\"uy\"uk olan ordinallerin en k\"u\c c\"u\u g\"ud\"ur.
\c Simdi yukar\i daki \eqref{eqn:1=0'} tan\i mlar\i n\i\ kullanabiliriz.
Ne s\i f\i r ne bir ard\i l olan ordinal, bir \textbf{limittir.}

\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

E\u ger $\{\xi\colon\xi<\alpha\}$ k\"umesi sonlu ise,
$\alpha$'ya da \textbf{sonlu} densin;
di\u ger durumda, \textbf{sonsuz.}
O zaman en k\"u\c c\"uk sonsuz ordinal bir limittir,
ve her limit ordinali sonsuzdur.
En k\"u\c c\"uk limit
\begin{equation*}
\upomega
\end{equation*}
olsun.
O zaman $\{\xi\colon\xi<\upomega\}$, do\u gal say\i lar k\"umesidir.

\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*}
\{\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}
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
\begin{equation*}
\{\xi\colon\xi<\alpha\}\included\bm A.
\end{equation*}
E\u ger $\alpha$ bir limit ise,
o zaman $\alpha$ 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, $\{\xi\colon\xi<\gamma\}\included\bm A$,
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{K\"ume aksiyomlar\i}\label{ch:ax}

Bu b\"ol\"umde $\on$'nin \sayfada{on-props}ki \"ozelliklerini
ve
\begin{equation}\label{eqn:pred}
\alpha=\{\xi\colon\xi<\alpha\}
\end{equation}
tan\i mlama imk\^an\i n\i,
k\"ume aksiyomlar\i ndan elde edece\u giz.
B\"ol\"umler \ref{ch:add}, \ref{ch:mul}, ve \numarada{ch:exp}
sadece \eqref{eqn:pred} e\c sitli\u gini kullanaca\u g\i z.

\section[Ordinaller varsa]{Ordinaller varsa}

Tan\i ma g\"ore
\begin{gather*}
	\emptyset=\{x\colon x\in x\land x\notin x\},\\
	a\cup\{b\}=\{x\colon x\in a\lor x=b\}.
\end{gather*}

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

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

\begin{axiom}[Yerle\c stirme]
Her $\bm F$ g\"ondermesi i\c cin,
e\u ger bir $A$ k\"umesinin her $b$ eleman\i\ i\c cin
$\bm F(b)$ tan\i mlan\i rsa, o zaman
\begin{equation*}
\{\bm F(x)\colon x\in a\}
\end{equation*}
s\i n\i f\i\ bir k\"umedir.
\end{axiom}

\begin{theorem}
  $\on$'de tan\i mlanan bir $\bm F$ g\"ondermesi i\c cin,
her $\alpha$ i\c cin
\begin{equation*}
  \bm F(\alpha)=\{\bm F(\xi)\colon\xi<\alpha\},
\end{equation*}
ve sonu\c c olarak, t\"um $\alpha$ ve $\beta$ i\c cin
\begin{equation}\label{eqn:a-f(a)}
\left.\qquad
  \begin{gathered}
  \alpha<\beta\liff\bm F(\alpha)\in\bm F(\beta),\\
  \alpha\leq\beta\liff\bm F(\alpha)\included\bm F(\beta).
\end{gathered}
\qquad\right\}
\end{equation}
\end{theorem}

\begin{proof}
Yerle\c stirme Aksiyomu sayesinde,
Ordinal \"Ozyineleme Teoreminin kan\i t\i n\i n y\"ontemini kullanarak
\begin{compactenum}[1)]
\item
$\bm F(0)=\emptyset$,
\item
$\bm F(\alpha')=\bm F(\alpha)\cup\{\bm F(\alpha)\}$,
\item
$\alpha$ limit ise
\begin{equation}\label{eqn:F(a)}
\bm F(\alpha)=\{\bm F(\xi)\colon\xi<\alpha\}
\end{equation}
\end{compactenum}
ko\c sullar\i n\i\ sa\u glayan bir $\bm F$ g\"ondermesinin var oldu\u gunu kan\i tlayabiliriz.
O zaman ordinal t\"umevar\i mdan her $\alpha$ i\c cin \eqref{eqn:F(a)} do\u grudur.
O halde
\begin{gather*}
	\alpha<\beta\lto\bm F(\alpha)\in\bm F(\beta),\\
	\alpha\leq\beta\lto\bm F(\alpha)\included\bm F(\beta).	
\end{gather*}
Tersleri de do\u grudur.  
Zira $\bm F(\alpha)\in\bm F(\beta)$ ama $\beta\leq\alpha$ ise,
o zaman
\begin{equation*}
\bm F(\alpha)\in\bm F(\alpha).
\end{equation*}
Bu durumda bir $\gamma$ i\c cin $\gamma<\alpha$ ve $\bm F(\alpha)=\bm F(\gamma)$,
dolay\i s\i yla $\bm F(\gamma)\in\bm F(\gamma)$.
O halde bo\c s olmayan $\{\xi\colon\bm F(\xi)\in\bm F(\xi)\}$ s\i n\i f\i n\i n en k\"u\c c\"uk eleman\i\ yoktur,
ki bu \c celi\c skidir.
Sonu\c c olarak
\eqref{eqn:a-f(a)} do\u grudur
\end{proof}

Her $\bm A$ s\i n\i f\i\ i\c cin
\begin{equation*}
\bigcup\bm A=\{x\colon\Exists y(y\in A\land x\in y)\}
\end{equation*}
olsun.  Bu yeni s\i n\i f, $\bm A$'n\i n \textbf{bile\c simidir.}
\"Ozel olarak
\begin{equation*}
B\cup C=\bigcup\{B,C\}.
\end{equation*}

\begin{corollary}
Teoremde $\bm F$ birebirdir,
dolay\i s\i yla $\bm F(\xi)=\xi$ varsay\i labilir.
Bu durumda
\begin{gather*}
  \alpha=\{\xi\colon\xi<\alpha\},\\
\alpha<\beta\liff\alpha\in\beta\liff\alpha\pincluded\beta.
\end{gather*}
Ayr\i ca $B\included\on$ ise
\begin{equation*}
  \sup B=\bigcup B.
\end{equation*}
\end{corollary}

\ksk

\c Simdi $\on$'nin sonu\c ctaki gibi oldu\u gunu varsayal\i m.
O zaman $\on$,
$\in$ ba\u g\i nt\i s\i\ taraf\i ndan iyis\i ralanm\i\c st\i r,
ve s\i n\i f\i n her eleman\i\ s\i n\i f\i n bir altk\"umesidir.
Ayr\i ca, s\i n\i f\i n her eleman\i n\i n ayn\i\ \"ozellikleri vard\i r
(yani eleman $\in$ ba\u g\i nt\i s\i\ taraf\i ndan iyis\i ralanm\i\c st\i r,
ve eleman\i n her eleman\i\ eleman\i n bir altk\"umesidir).

Herhangi $\bm A$ s\i n\i f\i\ i\c cin
\begin{equation*}
\Forall x(x\in\bm A\lto x\included\bm A)
\end{equation*}
ise, o zaman $\bm A$ \textbf{ge\c ci\c slidir.}
Ge\c ci\c sli k\"umeler
\begin{equation*}
\Forall y\bigl(y\in x\lto\Forall z(z\in y\lto z\in x)\bigr)
\end{equation*}
form\"ul\"u taraf\i ndan tan\i mlanm\i\c s s\i n\i f\i\ olu\c sturur.
Ayr\i ca $\in$ taraf\i ndan do\u grusal s\i ralanm\i\c s k\"umeler
\begin{multline*}
\Forall y(y\in x\lto y\notin y)\land{}\\
\Forall y\Forall z\Forall u\bigl(y\in x\land z\in x\land u\in x\lto{}\\
(y\in z\land z\in u\lto y\in u)\bigr)
\land{}\\
\Forall y\Forall z(y\in x\land z\in x\lto y\in z\lor y=z\lor y\in z)
\end{multline*}
form\"ul\"u taraf\i ndan tan\i mlanm\i\c s s\i n\i f\i\ olu\c sturur.
Bu form\"ul $\phi(x)$ ise
$\in$ taraf\i ndan iyis\i ralanm\i\c s k\"umeler
\begin{multline*}
	\phi(x)\land\Forall y\Bigl(\Forall z(z\in y\lto z\in x)\land\Exists zz\in y\lto{}\\
	\Exists z\bigl(z\in y\land\Forall u(u\in y\lto z\in u\lor z=u)\bigr)\Bigr)
\end{multline*}
form\"ul\"u taraf\i ndan tan\i mlanm\i\c s s\i n\i f\i\ olu\c sturur.

\section{Ordinaller vard\i r}

\c Simdi \sayfada{on-props}ki \"ozelliklerini unutunca,
yeniden her \textbf{ordinali,}\label{on-def}
\begin{compactenum}[1)]
\item
ge\c ci\c sli ve
\item
$\in$ taraf\i ndan iyis\i ralanm\i\c s
\end{compactenum}
bir k\"ume olarak tan\i mlar\i z.
O zaman ordinaller bir s\i n\i f olu\c stururlar.
\"Onceki gibi bu s\i n\i f $\on$ olsun,
ve $\on$'nin elemanlar\i,
k\"u\c c\"uk Yunan harfleriyle yaz\i ls\i n.
Sadece k\"ume aksiyomlar\i\ kullanarak,
\sayfada{on-props}ki \"ozellikleri
teorem olarak elde edece\u giz.

\begin{theorem}\label{thm:on-trans}
$\on$ ge\c ci\c slidir,
yani 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,
ve $d\in c$ 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,
yani $\on$'de $\in$, yans\i mas\i z ve ge\c ci\c slidir.
\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}

Her ordinalin bo\c s olmayan her $a$ altk\"umesinin,
$\in$ ba\u g\i nt\i s\i na g\"ore en k\"u\c c\"uk eleman\i d\i r.
Buradaki $a$'n\i n yerine bir \emph{s\i n\i f} kullanmak isteriz.
Sonraki aksiyomu kullanaca\u g\i z.

\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.

\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.

\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 $\alpha\pincluded\beta$.

\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.

\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 $\gamma\included\alpha$.

\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 $\alpha\included\gamma$.
\end{proof}

\c Simdi $\on$'nin $\in$ 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}

\c Simdi Teorem \ref{thm:on-trans} ve \numarada{thm:on-wo}n
$\on$ hem ge\c c\i\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.
Bu sonu\c c, Teorem \ref{thm:BFP} olarak g\"ord\"u\u g\"um\"uz
\textbf{Burali-Forti Paradoksudur.}

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

\ktk

\c Simdi
\begin{align*}
	0&=\emptyset,&
	\alpha'&=\alpha\cup\{\alpha\}
\end{align*}
tan\i mlayabiliriz.

\begin{axiom}[Sonsuzluk]
Ne limit olan ne limit i\c ceren ordinallerin olu\c sturdu\u gu s\i n\i f
bir k\"umedir.
\end{axiom}

Sonsuzluk Aksiyomu taraf\i ndan verilen k\"ume, $\upomega$'d\i r.

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

\begin{theorem}\sloppy
\Sayfada{on-def} tan\i mlanm\i\c s $\on$ s\i n\i f\i,
\Sayfada{on-props}ki \"ozellikleri sa\u glar.
\end{theorem}

\ktk

\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
$\begin{aligned}[t]
  \alpha+1
&=\alpha+0'&&\text{[\eqref{eqn:1=0'} tan\i m\i ndan]}\\
&=(\alpha+0)'&&\text{[\eqref{eqn:+'} tan\i m\i ndan]}\\
&=\alpha'.&&\text{[\eqref{eqn:+0} tan\i m\i ndan]}
\end{aligned}$\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)'&&\text{[\eqref{eqn:a'=min} tan\i m\i ndan]}\\
&=\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.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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  \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.)
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  \centering
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  \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.
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  \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.)
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(! 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)
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  \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.
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  \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}
\chapter{Old figures}

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%%%%%%%%%%%%%%%%%%%%%%%%%
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\psline(! 2 0.75 3 exp sub 0)(! 2 0.75 3 exp sub -0.03)
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\psline(! 2 0.75 5 exp sub 0)(! 2 0.75 5 exp sub -0.03)
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\psline(! 2 0.75 7 exp sub 0)(! 2 0.75 7 exp sub -0.03)
\psline(! 2 0.75 8 exp sub 0)(! 2 0.75 8 exp sub -0.03)
\psline(! 2 0.75 9 exp sub 0)(! 2 0.75 9 exp sub -0.03)
\psline(! 2 0.75 10 exp sub 0)(! 2 0.75 10 exp sub -0.03)
\psline(! 2 0.75 11 exp sub 0)(! 2 0.75 11 exp sub -0.03)
\psline(! 2 0.75 12 exp sub 0)(! 2 0.75 12 exp sub -0.03)
\psline(! 2 0.75 13 exp sub 0)(! 2 0.75 13 exp sub -0.03)
\psline(! 2 0.75 14 exp sub 0)(! 2 0.75 14 exp sub -0.03)
\psline(! 2 0.75 15 exp sub 0)(! 2 0.75 15 exp sub -0.03)
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%%%%%%%%%%%%%%%%%%%%%%
\psset{linestyle=dotted}
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  \end{pspicture}
  \caption{$\eta=\upomega+\xi$ denkleminin grafi\u gi}\label{fig:y=omega+x}
  
\end{figure}

\begin{figure}
  \centering
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%\psgrid
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%%%%%%%%%%%%%%%%%%%%%%
\uput[d](0,0){$0$}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%
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  \end{pspicture}

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


\begin{figure}
  \centering
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{pspicture}}

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


\begin{figure}
  \centering
\psset{unit=40mm,dotsize=6pt}
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%\psgrid
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%%%%%%%%%%%%%%%%%%%%%%
\uput[d](0,0){$1$}
\uput[d](0.25,0){$2$}
\uput[d](! 1 0.75 2 exp sub 0){$3$}
\uput[d](! 1 0.75 3 exp sub 0){$4$}
%\uput[d](0.578125,0){$3$}
\uput[d](1,0){$\upomega\vphantom1$}
\uput[d](1.25,0){$\upomega\cdot2$}
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%\uput[d](1.578125,0){$\upomega+3$}
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\uput[l](0,0.578125){$4$}
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\uput[l](0,1.4375){$\upomega\cdot3$}
\uput[l](0,1.578125){$\upomega\cdot4$}
\uput[l](0,2){$\upomega^2$}
\uput[l](0,2.25){$\upomega^2\cdot2$}
\uput[l](0,2.4375){$\upomega^2\cdot3$}
\uput[l](0,2.578125){$\upomega^2\cdot4$}
\uput[l](0,3){$\upomega^3$}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%
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\psline(! 2 0.75 7 exp sub 0)(! 2 0.75 7 exp sub -0.03)
\psline(! 2 0.75 8 exp sub 0)(! 2 0.75 8 exp sub -0.03)
\psline(! 2 0.75 9 exp sub 0)(! 2 0.75 9 exp sub -0.03)
\psline(! 2 0.75 10 exp sub 0)(! 2 0.75 10 exp sub -0.03)
\psline(! 2 0.75 11 exp sub 0)(! 2 0.75 11 exp sub -0.03)
\psline(! 2 0.75 12 exp sub 0)(! 2 0.75 12 exp sub -0.03)
\psline(! 2 0.75 13 exp sub 0)(! 2 0.75 13 exp sub -0.03)
\psline(! 2 0.75 14 exp sub 0)(! 2 0.75 14 exp sub -0.03)
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%%%%%%%%%%%%%%%%%%%%%%
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\psline(0,3)(2.1,3)
  \end{pspicture}
  \caption{$\eta=\upomega\cdot\xi$ denkleminin grafi\u gi}\label{fig:y=omega.x}
  
\end{figure}

\begin{figure}
  \centering
\psset{unit=40mm,dotsize=6pt}
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%\psgrid
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%%%%%%%%%%%%%%%%%%%%%%
\uput[d](0,0){$1$}
\uput[d](0.25,0){$2$}
\uput[d](! 1 0.75 2 exp sub 0){$3$}
\uput[d](! 1 0.75 3 exp sub 0){$4$}
%\uput[d](0.578125,0){$3$}
\uput[d](1,0){$\upomega\vphantom1$}
\uput[d](1.25,0){$\upomega\cdot2$}
%\uput[d](1.4375,0){$\upomega+2$}
%\uput[d](1.578125,0){$\upomega+3$}
\uput[d](2,0){$\upomega^2$}
\uput[l](0,0){$1$}
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\uput[l](0,0.4375){$3$}
\uput[l](0,0.578125){$4$}
\uput[l](0,1){$\upomega$}
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\uput[l](0,1.4375){$\upomega\cdot3$}
\uput[l](0,1.578125){$\upomega\cdot4$}
\uput[l](0,2){$\upomega^2$}
\uput[l](0,2.25){$\upomega^2\cdot2$}
\uput[l](0,2.4375){$\upomega^2\cdot3$}
\uput[l](0,2.578125){$\upomega^2\cdot4$}
\uput[l](0,3){$\upomega^3$}
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\uput[ul](2.2,0){$\xi$}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%
\psset{linestyle=dotted}
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\psline(0,3)(2.1,3)
\psdot[dotstyle=o](1,1)
\psdot[dotstyle=o](2,2)
  \end{pspicture}

  \caption{$\eta=\xi\cdot\upomega$ denkleminin grafi\u gi}\label{fig:y=x.omega}  
\end{figure}

%\end{document}


