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


\maketitle

\tableofcontents

%\listoffigures

%\mainmatter

\section{Ger\c cel Analiz}

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

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


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

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

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

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

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

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

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

\section{Ordinal say\i lar}

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

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

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

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

\begin{proof}
Her ordinalin daha b\"uy\"u\u g\"u oldu\u gundan
$\on$'nin \"usts\i n\i r\i\ yoktur.
$\on$'nin her altk\"umesinin \"usts\i n\i r\i\ oldu\u gundan
$\on$'nin kendisi k\"ume olamaz.
\end{proof}

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

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

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

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

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

Herhangi $\bm A$ s\i n\i f\i\ i\c cin
\begin{equation*}
\pow{\bm A},
\end{equation*}
$\bm A$'n\i n \emph{altk\"umeleri} taraf\i ndan olu\c sturulmu\c s s\i n\i ft\i r.

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

\c Simdi $\bm F\colon\on\to\on$ olsun.  E\u ger
\begin{compactenum}[1)]
\item
$\bm F$ \textbf{kesin artan,} yani
$\alpha<\beta\lto\bm F(\alpha)<\bm F(\beta)$, ve
\item
her $\alpha$ limiti i\c cin $\bm F(\alpha)=\sup\{\bm F(\xi)\colon\xi<\alpha\}$
\end{compactenum}
ise, o zaman $\bm F$'ye \textbf{normal} densin.

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

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

\subsection{Toplama}

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

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

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

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

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

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

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

\subsection{\c Carpma}

Tan\i m\i na g\"ore her $\alpha$ i\c cin
\begin{gather*}
	\alpha\cdot0=0,\\
	\alpha\cdot\beta'=\alpha\cdot\beta+\alpha,\\
	\gamma\text{ limit ise }\alpha\cdot\gamma=\sup\{\alpha\cdot\xi\colon\xi<\gamma\}.
\end{gather*}
\"Ozel olarak
\begin{equation*}
\alpha\cdot1=\alpha.
\end{equation*}
O zaman $\upomega\cdot2 
=\upomega\cdot1+\upomega
=\upomega+\upomega
=\sup\{\upomega+x\colon x\in\upomega\}$, ama
\begin{equation*}
2\cdot\upomega=\sup_{x\in\upomega}(2\cdot x)=\upomega,  
\end{equation*}
dolay\i s\i yla $2\cdot\upomega<\upomega\cdot2$.
\"Oyleyse \c carpma de\u gi\c smeli de\u gildir.

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

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

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

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

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

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

\subsection{Kuvvet alma}

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

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


\end{document}
