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\begin{document}
%\frontmatter
\title{Ordinal Analiz}
\subtitle{Aksiyomatik K\"umeler Kuram\i\ Dersi}
\author{David Pierce}
\date{24 Ocak 2020 tasla\u g\i}
\publishers{Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\.Istanbul\\
\url{dpierce@msgsu.edu.tr}\\
\url{mat.msgsu.edu.tr/~dpierce/}\\
\url{polytropy.com}}

\maketitle

%\frontmatter

%\selectlanguage{english}

%\mainmatter

%\selectlanguage{turkish}

\tableofcontents

\listoffigures

%\mainmatter


\chapter{Harfler ve Aksiyomlar}%\label{ch:let}

\section{Harfler}

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

%  \begin{list}{#1}{\setlength{\leftmargin}{2mm}}
\begin{description}
\item[K\"u\c c\"uk Latin harfleri]\mbox{}
 \begin{itemize}[]
\item
  $a$, $b$, $c$, $d$, $e$\letsep k\"umeler
  \letterrefs{sayfa \pageref{abc}}
\item
  $f$, $g$, $h$\letsep tan\i m k\"umesi olan g\"ondermeler
  \letterrefs{sayfa \pageref{thm:rec}}
\item
$i$, $j$\letsep do\u gal say\i\ de\u gi\c skenleri
  \letterrefs{sayfa \pageref{i}}
\item
  $k$, $\ell$, $m$, $n$\letsep do\u gal say\i lar
  \letterrefs{sayfa \pageref{k}}
\item
  $s$, $t$\letsep terimler
  \letterrefs{sayfa \pageref{t}}
\item
  $u$, $v$, $w$, $x$, $y$, $z$\letsep k\"ume de\u gi\c skenleri
  \letterrefs{sayfa \pageref{xyz}}
\end{itemize}
\item[B\"uy\"uk Latin harfleri]\mbox{}
\begin{itemize}[]
\item
  $A$, $B$, $C$, $D$\letsep k\"umeler
  \letterrefs{sayfa \pageref{A-harfi}}
\item
$X$, $Y$, $Z$\letsep k\"ume de\u gi\c skenleri
  \letterrefs{sayfa \pageref{X}}
\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{sayfa \pageref{curly-C}}
\item
  $\pow A$\letsep$\{X\colon X\included A\}$
  \letterrefs{sayfa \pageref{pow} ve \pageref{pow-2}}
\item
  $\powf A$\letsep$\{X\in\pow A\colon\card X<\upomega\}$
  \letterrefs{sayfa \pageref{powf}}
\end{itemize}
\item[D\"uz b\"uy\"uk Latin harfleri]\mbox{}
  \begin{itemize}[]
  \item
    $\mathsf A$, $\mathsf B$\letsep ifadeler
    \letterrefs{sayfa \pageref{ifade}}
  \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{sayfa \pageref{bm-A}}
\item
  $\bm F$, $\bm G$, $\bm H$\letsep
  %s\i n\i fta tan\i mlanm\i\c s
  g\"ondermeler
\letterrefs{sayfa \pageref{bm-F}}
\end{itemize}
\item[Dikey b\"uy\"uk siyah Latin harfleri]\mbox{}
\begin{itemize}[]
\item
$\cn$\letsep kardinaller s\i n\i f\i\
  \letterrefs{sayfa \pageref{CN}}
\item
$\on$\letsep ordinaller s\i n\i f\i\
  \letterrefs{sayfa \pageref{ON}}
\item
  $\universe$\letsep evrensel s\i n\i f
  \letterrefs{sayfa \pageref{V}}
\end{itemize}
\item[Yunan harfleri]\mbox{}
\begin{itemize}[]
\item
  $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$\letsep
  ordinaller
  \letterrefs{sayfa \pageref{alpha}}
\item
  $\zeta$, $\eta$\letsep ordinal de\u gi\c skenler
  \letterrefs{sayfa \pageref{xi}}
\item
  $\theta$\letsep ordinal
  \letterrefs{sayfa \pageref{theta}}
\item
  $\kappa$, $\lambda$, $\mu$, $\nu$\letsep kardinaller
  \letterrefs{sayfa \pageref{kappa} ve \pageref{mu}}
\item
  $\xi$\letsep ordinal de\u gi\c sken
  \letterrefs{sayfa \pageref{xi}}
\item
  $\pi$\letsep form\"ul
  \letterrefs{sayfa \pageref{pi}}
\item
  $\rho$, $\sigma$, $\tau$\letsep c\"umleler
  \letterrefs{sayfa \pageref{sigma}}
\item
  $\phi$, $\chi$, $\psi$\letsep form\"uller
  \letterrefs{sayfa \pageref{phi}, \pageref{chi}, \pageref{psi}}
\end{itemize}
\item[Dikey Yunan harfi]\mbox{}
\begin{itemize}[]
\item
  $\upvarepsilon_0$\letsep
  $\sup\{\upomega,\upomega^{\upomega},\upomega^{\upomega^{\upomega}},\dots\}$
\letterrefs{sayfa \pageref{epsilon}}
\item
$\upomega$\letsep do\u gal say\i lar\i\ k\"umesi \letterrefs{sayfa \pageref{omega}}
\end{itemize}
\item[Dikey k\"u\c c\"uk Latin harfleri]\mbox{}
\begin{itemize}[]
\item
  $\cardinal$\letsep kardinal
  \letterrefs{sayfa \pageref{card}}
\item
$\cofinality$\letsep kofinallik
  \letterrefs{sayfa \pageref{cof}}
\item
  $\max$\letsep en b\"uy\"uk
  \letterrefs{sayfa \pageref{max}}
\item
$\min$\letsep en k\"u\c c\"uk
  \letterrefs{sayfa \pageref{min}}
\item
  $\sup$\letsep en k\"u\c c\"uk \"usts\i n\i r
  \letterrefs{sayfa \pageref{sup}}
\end{itemize}
\item[Dikey b\"uy\"uk Latin harfleri]\mbox{}
\begin{itemize}[]
\item
  \gch\letsep Genelle\c stirilmi\c s Kontin\"u Hipotezi
  \letterrefs{sayfa \pageref{GCH}}
\item
  \ch\letsep Kontin\"u Hipotezi
  \letterrefs{sayfa \pageref{CH}}
\item
  \zf\letsep Zermelo--Fraenkel Aksiyomlar\i\
  \letterrefs{sayfa \pageref{ZF}}
\item
  \zfc\letsep Se\c cim Aksiyomu ile {\zf}
  \letterrefs{sayfa \pageref{ZFC}}
\end{itemize}
\item[``Tahta siyah\i'' harfleri]\mbox{}
\begin{itemize}[]
\item
  $\N$\letsep sayma say\i lar\i\ k\"umesi
  \letterrefs{sayfa \pageref{N}}
\item
$\Q$\letsep kesirli say\i lar k\"umesi \letterrefs{sayfa \pageref{Q}}
\item
$\R$\letsep ger\c cel say\i lar k\"umesi \letterrefs{sayfa \pageref{R}}
\item
$\Z$\letsep tamsay\i lar k\"umesi \letterrefs{sayfa \pageref{Z}}
\end{itemize}
\item[Harflerden t\"ureyen simgeler]\mbox{}
\begin{itemize}[]
\item
  $\forall$\letsep her \lips i\c cin (\emph{for \textbf All};
  sayfa \pageref{forall})
\item
$\exists$\letsep baz\i\lips i\c cin (\emph{there \textbf Exists};
  sayfa \pageref{exists})
\item
$\in$\letsep %eleman olma ba\u g\i nt\i s\i\ 
(``$a\in B$'' = ``$a$ \gr{>est`i} $B$''; sayfa \pageref{in})
\item
$\cup$, $\bigcup$\letsep bile\c sim (\emph{\textbf Union};
  sayfa \pageref{union} ve \pageref{Union})
\item
$\lor$\letsep veya (Latince \textsc{\textbf Vel};
  sayfa \pageref{vel})
\end{itemize}
\end{description}

\section{Aksiyomlar}

K\"umeler i\c cin on tane aksiyom vard\i r.

\begin{description}
\item[E\c sitlik]
  (sayfa \pageref{ax:equality}):
  e\c sit k\"umeler, ayn\i\ k\"umelerin elemanlar\i d\i r.
\item[Ay\i rma]
  (sayfa \pageref{ax:separation}):
  her k\"ume her alts\i n\i f\i, bir k\"umedir.
\item[Bo\c s K\"ume]
  (sayfa \pageref{ax:empty-set}):
  bo\c s s\i n\i f, bir k\"umedir.
\item[Biti\c stirme]
  (sayfa \pageref{ax:adjunction}):
  bir k\"umeden bir eleman\i\ daha olan s\i n\i f, bir k\"umedir.
\item[Sonsuzluk]
  (sayfa \pageref{ax:infinity}):
  do\u gal say\i lar s\i n\i f\i, bir k\"umedir.
\item[Yerle\c stirme]
  (sayfa \pageref{ax:replacement}):
  bir g\"onderme alt\i nda bir k\"umenin imgesi, bir k\"umedir.
\item[Bile\c sim]
  (sayfa \pageref{ax:union}):
    bir k\"umenin bile\c simi, bir k\"umedir.
\item[Kuvvet K\"umesi]
  (sayfa \pageref{ax:power-set}):
  bir k\"umenin altk\"umelerinin olu\c sturdu\u gu s\i n\i f,
  bir k\"umedir.
\item[Temel]
  (kullan\i lmam\i\c s):
 her bo\c s olmayan k\"umenin,
kendisinden ayr\i k bir eleman\i\ vard\i r.
\item[Se\c cim]
  (sayfa \pageref{ax:choice}):
  her k\"ume iyis\i ralanabilir.
\end{description}

Yukar\i daki ilk dokuz aksiyomlar\i,
\textbf{Zermelo--Fraenkel Aksiyomlar\i d\i r,}
k\i saca {\zf}.
Hep beraber, aksiyomlara {\zfc} denir.

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

\section{K\"umeler ve S\i n\i flar}

\begin{compactitem}
  \item
  \"Oklid'in \emph{\"O\u geler}'inde ortak kavramlar{\i} kullanarak
  postulatlardan di\u ger \"onermeleri kan{\i}tlar{\i}z.
\item
  K\"umeler kuram{\i}nda mant{\i}\u g{\i}n kurallar{\i}n{\i} kullanarak
  aksiyomlardan di\u ger teoremleri kan{\i}tlar{\i}z.
\end{compactitem}
K\"umeler kuram\i nda her teorem,
\emph{serbest de\u gi\c skeni olmayan bir form\"uld\"ur,}
k\i saca bir \textbf{c\"umledir}
(\eng{sentence}).

Tek serbest de\u gi\c skeni olan bir form\"ulde,
form\"ul\"un serbest de\u gi\c skeninin
\emph{serbest ge\c ci\c slerinin} yerine
bir \emph{k\"umenin} ad\i\ konulabilir.
E\u ger form\"ul \label{phi}$\phi$ ve k\"ume $a$ ise,
o zaman meydana gelen c\"umle
\begin{equation*}
  \fbox{$\phi(a)$}
\end{equation*}
olur.
ya \emph{do\u grudur} ya da \emph{yanl\i\c st\i r.}
E\u ger do\u gru ise,
o zaman $a$ k\"umesi, $\phi$ form\"ul\"un\"u \textbf{sa\u glar}
(\eng{satisfies}).

Bir form\"ul\"u sa\u glayan k\"umeler,
bir \textbf{s\i n\i f} olu\c sturur.
Olu\c sturan k\"umeler, s\i n\i f\i n \textbf{elemanlar\i d\i r}
(\eng{elements}),
ve form\"ul, s\i n\i f\i\ \textbf{tan\i mlar}
(\eng{defines}).
E\u ger form\"ul $\phi$
ve tek serbest de\u gi\c skeni $x$ ise,
o zaman
\begin{equation*}
  \fbox{$\{x\colon\phi\}$}
\end{equation*}
s\i n\i f\i\ tan\i mlan\i r.

K\"umelerin de elemanlar\i\ vard\i r,
ve her biri bir k\"umedir.
E\u ger bir $a$ k\"umesi, bir $b$ k\"umesinin bir eleman\i\ ise,
o zaman
\begin{equation*}\label{in}
  \fbox{$a\in b$}
\end{equation*}
c\"umlesi yaz\i l\i r.
Aksi durumda
\begin{equation*}
  \fbox{$a\notin b$}
\end{equation*}
yaz\i l\i r.

\begin{theorem}
  Her k\"ume, bir s\i n\i ft\i r.
\end{theorem}

\begin{proof}
Her $a$ k\"umesi, $\{x\colon x\in a\}$ s\i n\i f\i d\i r.
\end{proof}

\begin{theorem}[Russell Paradoksu]\label{thm:Russell}
  Baz\i\ s\i n\i f, k\"ume de\u gildir.
\end{theorem}

\begin{proof}
  Her $a$ k\"umesi i\c cin a\c sa\u g\i daki iki ko\c sul denktir.
  \begin{compactenum}
    \item
      $a$, $\{x\colon x\notin x\}$ s\i n\i f\i n\i n eleman\i d\i r.
    \item
      $a$, $a$ k\"umesinin eleman\i\ de\u gildir (yani $a\notin a$).
  \end{compactenum}
  \"Oyleyse her k\"ume,
  $\{x\colon x\notin x\}$ s\i n\i f\i ndan
  farkl\i d\i r.
\end{proof}

\section{Form\"uller}

Form\"ullerde kullanaca\u g\i m\i z simgeler
ya \emph{mant\i ksald\i r} ya da \emph{mant\i ksal olmayand\i r.}
\begin{compactitem}
\item
\textbf{Mant\i ksal} (\eng{logical}) simgeler:
\begin{compactitem}
\item
  \textbf{terimler}\index{terim} (\eng{terms}):
  \begin{compactitem}
\item\label{abc}
  \textbf{sabitler}\index{sabit} (\eng{constants}):
  \fbox{$a$, $b$, $c$, \dots}%
\glossary{$a$, $b$, $c$, \dots}
\item\label{xyz}
\textbf{de\u gi\c skenler}\index{de\u gi\c sken} (\eng{variables}):
\fbox{$z$, $y$, $x$, \dots}%
\glossary{$x$, $y$, $z$, \dots}
  \end{compactitem}
\item
  \textbf{ba\u glay\i c\i lar}\index{ba\u glay\i c\i} (\eng{connectives}):
  \begin{compactitem}
    \item
\textbf{iki-konumlu} (\eng{binary}) bir ba\u glay\i c\i:
  \fbox{$\land$}
\item
  \textbf{tek-konumlu} (\eng{singulary}) bir ba\u glay\i c\i: \fbox{$\lnot$}
  \end{compactitem}
\item\label{exists}
  bir \textbf{niceleyici}\index{niceleyici} (\eng{quantifier}):
  \fbox{$\exists$}
\item
  \textbf{ayra\c clar}\index{ayra\c c} (\eng{parentheses, brackets}):
  \begin{compactitem}
  \item
    sol: \fbox{$($}
  \item
    sa\u g: \fbox{$)$}
  \end{compactitem}
\end{compactitem}
\item
\textbf{mant\i ksal olmayan} (\eng{non-logical}) bir simge:
\begin{compactitem}
  \item
iki-konumlu bir \textbf{y\"uklem}\index{y\"uklem} (\eng{predicate}):
\fbox{$\in$}
\end{compactitem}
\end{compactitem}
\index{form\"ul}\textbf{Form\"uller} (\eng{formulas}) i\c cin
tan\i m \emph{\"ozyinelidir.}
Asl\i nda form\"ullerin d\"ort tane t\"ur\"u vard\i r.
\begin{compactenum}
\item
\label{t}$t$ ve $s$ terim olmak \"uzere
\begin{equation*}
\fbox{$t\in s$}
\end{equation*}
bir \textbf{i\c cermedir} (\eng{containment}).  
\item
$\phi$ bir form\"ul olmak \"uzere
\begin{equation*}
\fbox{$\lnot\phi$}
\end{equation*}
bir \textbf{de\u gillemedir}%
\index{de\u gilleme}
(\eng{negation}).
\item
  $\phi$ ve
  \label{psi}$\psi$ form\"ul olmak \"uzere
  \begin{equation*}
\fbox{$(\phi\land\psi)$}
  \end{equation*}
bir \textbf{t\"umel evetlemedir}
\index{t\"umel evetleme}%
\index{evetleme}
(\eng{conjunction}).
\item
$\phi$ bir form\"ul,  $x$ bir de\u gi\c sken olmak \"uzere
  \begin{equation*}
\fbox{$\Exists x\phi$}
  \end{equation*}
bir \textbf{\"orneklemedir}%
\index{\"ornekleme}
(\eng{instantiation}). 
\end{compactenum}
Her i\c cerme,
her de\u gilleme,
her t\"umel evetleme,
ve her \"ornekleme,
bir form\"uld\"ur.
Ayr\i ca
\begin{compactitem}
  \item
$\phi$,
hem $\lnot\phi$ de\u gillemesinin
hem de $\Exists x\phi$ \"orneklemesinin tek \textbf{bile\c senidir}
(\eng{component});
\item
  $\phi$ ve $\psi$,
  $(\phi\land\psi)$ \"orneklemesinin bile\c senleridir.
\end{compactitem}

\begin{lemma}\label{lem:pis}
  E\u ger simgeler, bir form\"ul\"un
sonuna eklenirse
veya
  sonundan kald\i r\i l\i rsa,
  o zaman
yeni bir form\"ul elde edilmez.
\end{lemma}

\begin{proof}
  Form\"ullerin tan\i m\i n\i n sa\u glad\i\u g\i\
  t\"umevar\i m y\"ontemini kullanaca\u g\i z.
\begin{asparaenum}
\item
  Bir i\c cerme i\c cin 
  iddiam\i z do\u grudur,
  \c c\"unk\"u
  i\c cermeler ve sadece i\c cermeler,
  terim ile ba\c slar,
  ve bu form\"ullerin her birinin uzunlu\u gu \"u\c ct\"ur.
\item
  T\"umevar\i m hipotezi olarak
  iddiam\i z\i n bir $\phi$ form\"ul\"u i\c cin do\u gru olmas\i\
  kabul edilsin.
  M\"umk\"unse simgeler $\lnot\phi$ form\"ul\"un\"un
sonuna ekleyerek
veya
sonundan kald\i rarak
yeni bir form\"ul elde edilsin.
o zaman ortaya \c c\i kan form\"ul,
bir \label{ifade}$\mathsf A$ ifadesi i\c cin,
$\lnot\mathsf A$ bi\c cimindedir.
Bu durumda $\mathsf A$,
bir $\psi$ form\"ul\"u olmal\i d\i r,
ve simgeler $\phi$'nin sonuna ekleyerek
veya
sonundan kald\i rarak
$\psi$ elde edilir, ki bu
hipoteze g\"ore imk\^ans\i zd\i r.
O zaman iddiam\i z $\lnot\phi$ i\c cin do\u grudur.
\item
  T\"umevar\i m hipotezi olarak
  iddiam\i z\i n bir $\phi$ form\"ul\"u ve bir $\psi$ form\"ul\"u i\c cin
  do\u gru olmas\i\
  kabul edilsin.
  M\"umk\"unse simgeler $(\phi\land\psi)$ form\"ul\"un\"un
sonuna ekleyerek
veya
sonundan kald\i rarak
yeni bir form\"ul elde edilsin.
O zaman ortaya \c c\i kan form\"ul,
bir $\mathsf A$ ifadesi i\c cin,
$(\mathsf A$ bi\c cimindedir.
Bu durumda $\mathsf A$,
bir
\label{chi}$\chi$ ve bir \label{pi}$\pi$ form\"ul\"u i\c cin
$\chi\land\pi)$ olmal\i d\i r.
Bu durumda
$\chi$, $\phi$ olmal\i d\i r,
\c c\"unk\"u aksi durumda
simgeler
$\phi$'nin
sonuna ekleyerek
veya
sonundan kald\i rarak
$\chi$ elde edilir,
ve hipoteze g\"ore bu imk\^ans\i zd\i r.
\c Simdi ayn\i\ \c sekilde
$\pi$, $\psi$ olmal\i d\i r.
  O zaman iddiam\i z $(\phi\land\psi)$ i\c cin do\u grudur.
\item
  Benzer \c sekilde
  iddiam\i z\i n bir $\phi$ form\"ul\"u i\c cin do\u gru olmas\i\
  kabul edilirse,
  o zaman $\Exists x\phi$ i\c cin de do\u grudur.
\end{asparaenum}

B\"oylece t\"umevar\i mdan her durumda iddiam\i z do\u grudur.
\end{proof}

\begin{theorem}[Tek Okunabilirlik]\mbox{}\label{thm:ur}
  \begin{compactenum}
    \item
      Her form\"ul\"un tek bir t\"ur\"u vard\i r.
    \item
      Tek bir \c sekilde bir form\"ul
      t\"umel evetleme olabilir.
  \end{compactenum}
\end{theorem}

\"Orne\u gin $\bigl(\phi\land(\psi\land\chi)\bigr)$ form\"ul\"u,
t\"umel evetlemedir;
ayr\i ca
tek bir \c sekilde t\"umel evetlemedir,
\c c\"unk\"u
sadece $\phi$ ve $(\psi\land\chi)$,
verilen form\"ul\"un bile\c senleridir.
Asl\i nda
$\phi\land(\psi$ ve $\chi)$
s\i ras\i yla $\mathsf A$ ve $\mathsf B$ ile
g\"osterilirse,
o zaman
$\bigl(\phi\land(\psi\land\chi)\bigr)$,
$(\mathsf A\land \mathsf B)$ olur,
ama $\mathsf A$ ile $\mathsf B$
bu form\"ul\"un bile\c senleri de\u gildir,
\c c\"unk\"u form\"ul de\u gildir. 

\begin{proof}[Teoremin kan\i t\i.]
  Bir form\"ul\"un ilk simgesi
  \begin{compactitem}
  \item
    bir terim ise, form\"ul bir i\c cermedir;
  \item
    $\lnot$ ise, form\"ul bir de\u gillemedir;
  \item
    $($ ise, form\"ul bir t\"umel evetlemedir;
  \item
    $\exists$ ise, form\"ul bir \"orneklemedir.
  \end{compactitem}
  Ayr\i ca e\u ger
  $\phi$, $\psi$, $\chi$, ve $\pi$ form\"ulleri i\c cin
$(\phi\land\psi)$ ve $(\chi\land\pi)$
  ayn\i\ form\"ul ise,
  o zaman
  Lemma \numarada{lem:pis}n $\phi$ ve $\chi$ ayn\i\ form\"uld\"ur,
  dolay\i s\i yla $\psi$ ve $\pi$ ayn\i\ form\"uld\"ur.
\end{proof}

  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.
\"Orne\u gin $\Exists x(x\in y\land x\in z)$
form\"ul\"unde $x$ de\u gi\c skeninin \"u\c c tane ge\c ci\c si vard\i r
(ayr\i ca $y$ ve $z$'nin birer ge\c ci\c si vard\i r).

\Teorem{thm:ur} sayesinde
bir form\"ulde bir de\u gi\c skenin bir ge\c ci\c sinin
\textbf{serbest} (\eng{free})
olmas\i na
\emph{\"ozyineli} bir tan\i m verebiliriz:
\begin{compactenum}
\item
Bir i\c cermede bir de\u gi\c skenin her ge\c ci\c si,
serbesttir.
\item
  Bir de\u gillemenin bile\c seninde
    bir de\u gi\c skenin
serbest bir ge\c ci\c si,
   de\u gillemenin kendisinde serbesttir.
\item
  Bir t\"umel evetlemenin bir bile\c seninde
    bir de\u gi\c skenin
serbest bir ge\c ci\c si,
  t\"umel evetlemenin kendisinde serbesttir.
\item
  Bir $\Exists x\phi$ \"orneklemesinde
  \begin{compactitem}
  \item
    $x$'ten farkl\i\ olan bir de\u gi\c skenin
  $\phi$'deki serbest bir ge\c ci\c si
    %\"orneklemenin kendisinde
    serbesttir;
  \item
    $x$'in hi\c c serbest ge\c ci\c si yoktur.
  \end{compactitem}
\end{compactenum}
Serbest olmayan bir ge\c ci\c s, \textbf{ba\u gl\i d\i r}
(\eng{bound}).

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
\label{sigma}$\sigma$, $\tau$, ve $\rho$ gibi
Yunan harflerini kullanaca\u g\i z.

\c Simdilik $\phi$, tek serbest de\u gi\c skeni $x$ olan bir form\"ul;
$a$, bir k\"ume;
ve $y$, bir de\u gi\c sken olsun.
E\u ger $x$'in $\phi$'deki her serbest ge\c ci\c sinin yerine
$a$ konulursa, o zaman g\"ord\"u\u g\"um\"uz gibi
$\phi(a)$ c\"umesi elde edilir.
E\u ger $\phi$'de
$y$'in ba\u gl\i\ oldu\u gu yerde $x$ serbest de\u gilse,
ve $x$'in $\phi$'deki her serbest ge\c ci\c sinin yerine
$y$ konulursa, o zaman
\begin{equation*}
  \fbox{$\phi(y)$}
\end{equation*}
form\"ul\"u elde edilir.
\"Orne\u gin $\phi$, $\Exists yy\in x$ ise,
o zaman $y$, verilen ko\c sulu sa\u glamaz,
ama $\phi(z)$, $\Exists yy\in z$ olur.

E\u ger $\phi$'nin birden fazla serbest de\u gi\c skeni varsa,
o zaman benzer ifadeler kullan\i labilir
(\"orne\u gin \Teoremin{thm:sub} kan\i t\i nda).


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

\"Ozyine bir tan\i ma g\"ore
her c\"umle ya
\textbf{do\u grudur}%
\index{do\u gru}\label{truth}
(\eng{true}) ya da
\textbf{yanl\i\c st\i r}%
\index{yanl\i\c s}
(\eng{false}).
\begin{compactenum}
\item
  Bir $a$ k\"umesi bir $b$ k\"umesinin bir eleman\i\ ise,
  o zaman $a\in b$ i\c cermesi do\u grudur;
  aksi durumda yanl\i\c st\i r.
\item
  Bile\c seni do\u gru ise,
  o zaman bir de\u gilleme yanl\i\c st\i r;
    aksi durumda do\u grudur.
  \item
    Bile\c senlerinin her biri do\u gru ise,
    bir t\"umel evetleme de do\u grudur;
    aksi durumda yanl\i\c st\i r. 
\item
  Bir $a$ i\c cin $\phi(a)$ c\"umlesi do\u gru ise,
  o zaman $\Exists x\phi(x)$ \"orneklemesi de do\u grudur;
    aksi durumda yanl\i\c st\i r. 
\end{compactenum}

\section{K\i saltmalar}

Baz\i\ form\"uller k\i salabilir.

\subsection{Mant\i k}

Yeni iki-konumlu ba\u glay\i c\i lar ile
\begin{compactitem}
\item
  $\lnot(\lnot\phi\land\lnot\psi)$ form\"ul\"u
  \begin{equation*}\label{vel}
    \fbox{$(\phi\lor\psi)$}
  \end{equation*}
\textbf{tikel evetlemesi} 
(\eng{disjunction}),
\item
  $(\lnot\phi\lor\psi)$ form\"ul\"u
  \begin{equation*}
    \fbox{$(\phi\lto\psi)$}
  \end{equation*}
\textbf{gerektirmesi}%
\index{gerektirme}
(\eng{implication}),
\item
  $\bigl((\phi\lto\psi)\land(\psi\lto\phi)\bigr)$ form\"ul\"u
  \begin{equation*}
    \fbox{$(\phi\liff\psi)$}
  \end{equation*}
\textbf{denkli\u gi}%
\index{denklik}
(\eng{equivalence})
\end{compactitem}
  olarak k\i saca yaz\i l\i r.
  Ayr\i ca
  yeni bir niceleyici ile
  \begin{compactitem}
    \item
  $\lnot\Exists x\lnot\phi$ form\"ul\"u
  \begin{equation*}\label{forall}
    \fbox{$\Forall x\phi$}
  \end{equation*}
\textbf{genelle\c stirmesi}%
\index{genelle\c stirme}
(\eng{generalization})
  \end{compactitem}
  olarak k\i saca yaz\i l\i r.
  

  \subsection{Ayra\c clar}
  
\begin{compactitem}
\item
Varsa, bir form\"ul\"un en d\i\c staki ayra\c clar\i\ yaz\i lmayabilir.
\item
$\lto$ ve $\liff$ ba\u glay\i c\i lar\i na g\"ore $\land$ ve
  $\lor$ ba\u glay\i c\i lar\i na \"onceli\u gi verilebilir:
  \"orne\u gin
$\phi\land\psi\lto\chi$ 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{compactitem}

\subsection{Form\"uller}

Bir y\"uklemin \textbf{yorumu}
(\eng{interpretation}),
bir \textbf{ba\u g\i nt\i d\i r}
(\eng{relation}).
\"Orne\u gin $\in$ y\"ukleminin yorumu,
\textbf{eleman olma}
(\eng{being an element})
ba\u g\i nt\i s\i d\i r.
Bu ba\u g\i nt\i, $x\in y$ form\"ul\"u taraf\i ndan tan\i mlan\i r.
Ba\c ska form\"ullerin tan\i mlad\i\u g\i\ ba\u g\i nt\i lar i\c cin
yeni y\"uklemler kullanaca\u g\i z.

Bir $\Forall x(x\in t\lto x\in s)$ form\"ul\"un\"un yerine
\begin{equation*}
  \fbox{$t\included s$}
\end{equation*}
\textbf{kapsamas\i} (\eng{inclusion})
yaz\i l\i r.
E\u ger $a\included b$ ise
(yani c\"umle do\u gru ise),
o zaman $a$, $b$'nin bir \textbf{altk\"umesidir}
(\eng{subset}).

Bir $(t\included s\land s\included t)$ form\"ul\"un\"un yerine
\begin{equation*}
  \fbox{$t= s$}
\end{equation*}
\textbf{e\c sitli\u gi}
(\eng{equation})
yaz\i l\i r.
E\u ger $a= b$ ise,
o zaman $a$, $b$'ye \textbf{e\c sittir}
(\eng{equal}).

\begin{theorem}\label{thm:=}
  Her $a$ ve her $b$ k\"umesi i\c cin
  \begin{equation*}
    a=b\liff\Forall x(x\in a\liff x\in b).
  \end{equation*}
\end{theorem}

\ktk

\.I\c cermelerin,
kapsamalar\i n,
ve e\c sitliklerin
\"ozel de\u gillemeleri vard\i r.
\begin{equation*}
  \begin{array}{cc}
    \text{form\"ul}&\text{de\u gillemesi}\\\hline
    t\in s&t\notin s\\
    t\included s&t\nincluded s\\
    t=s&t\neq s\\\hline
  \end{array}
\end{equation*}

Bir $(t\included s\land t\neq s)$ form\"ul\"un\"un yerine
\begin{equation*}
  \fbox{$t\pincluded s$}
\end{equation*}
yaz\i l\i r; tabii ki form\"ul\"un de\u gillemesi
$t\npincluded s$.
E\u ger $a\pincluded b$ ise,
o zaman $a$, $b$'nin bir \textbf{\"ozaltk\"umesidir}
(\eng{proper subset}).


\c Simdi $\phi$ ve $\psi$,
tek serbest de\u gi\c skeni $x$ olan form\"ul olsun.
Ayr\i ca
$\phi$ ve $\psi$ taraf\i ndan tan\i mlanm\i\c s s\i n\i flar
s\i ras\i yla
\label{bm-A}$\bm A$ ve $\bm B$ olsun.
O zaman
$\Forall x(\phi\lto\psi)$ form\"ul\"un\"un yerine
\begin{equation*}
  \fbox{$\bm A\included\bm B$}
\end{equation*}
kapsamas\i\ yaz\i labilir, ve
$\bm A\included\bm B\land\bm B\included\bm A$
form\"ul\"un\"un yerine
\begin{equation*}
  \fbox{$\bm A=\bm B$}
\end{equation*}
e\c sitli\u gi yaz\i l\i r.
Bir form\"ulde
k\"ume olmayan bir s\i n\i f
bir k\"umenin yerini alamaz, ama
$\Exists x(x=\bm A\land\psi)$
form\"ul\"un\"un yerine
\begin{equation*}
  \fbox{$\psi(\bm A)$}
\end{equation*}
k\i saltmas\i\ yaz\i labilir.

\subsection{S\i n\i flar}

Baz\i\ s\i n\i flar\i n k\i saltmas\i\ vard\i r.
A\c sa\u g\i da $\bm A$ ve $\bm B$,
rastgele k\"umedir.
\label{union}\label{V}\label{pow}\begin{equation*}
  \begin{array}{ccc}
    \text{form\"ul}
    &\parbox{24mm}{\centering
      tan\i mlanm\i\c s\\s\i n\i f\vphantom p}
    &\text{ad\i}\\\hline
    x\in \bm A\land x\in \bm B&\bm A\cap \bm B&\text{$\bm A$ ve $\bm B$'nin
      \textbf{kesi\c simi}}\\
    x\in \bm A\lor x\in \bm B&\bm A\cup \bm B&\text{$\bm A$ ve $\bm B$'nin
      \textbf{birle\c simi}}\\
    x\in \bm A\land x\notin \bm B&\bm A\setminus \bm B
    &\text{$\bm A$'n\i n $\bm B$'den \textbf{fark\i}}\\
    x=x&\universe&\textbf{evrensel s\i n\i f}\\
    x\neq x&\emptyset&\textbf{bo\c s s\i n\i f}\\
    x\included\bm A&\pow{\bm A}
    &\text{$\bm A$'n\i n \textbf{kuvvet s\i n\i f\i}}\\
    \Forall y(y\in\bm A\lto x\in y)&\bigcap\bm A
    &\text{$\bm A$'n\i n \textbf{kesi\c simi}}\\\label{Union}
    \Exists y(y\in\bm A\land x\in y)&\bigcup\bm A
    &\text{$\bm A$'n\i n \textbf{birle\c simi}}\\
    %x\in\bm A\land\phi&\{x\in\bm A\colon\phi\}&\\
    \hline
  \end{array}
\end{equation*}
Tek serbest de\u gi\c skeni $x$ olan bir $\phi$ form\"ul\"u i\c cin
$x\in\bm A\land\phi$ form\"ul\"u
\begin{equation*}
  \fbox{$\{x\in\bm A\colon\phi\}$}
\end{equation*}
s\i n\i f\i n\i\ tan\i mlar.

A\c sa\u g\i da $a$ ve $b$,
rastgele k\"umedir.
\begin{equation*}
  \begin{array}{ccc}
    \text{form\"ul}
    &\text{tan\i mlanm\i\c s s\i n\i f}
    &\text{ad\i}\\\hline
    x\in a\lor x=a&a'&\text{$a$'n\i n \textbf{ard\i l\i}}\\
    x=a&\{a\}&\\
    x=a\lor x=b&\{a,b\}&
  \end{array}
\end{equation*}


\section{E\c sitlik}

\Teoreme{thm:=} g\"ore e\c sit k\"umelerin elemanlar\i\ ayn\i d\i r.

\begin{axiom}[E\c sitlik]\label{ax:equality}
  E\c sit k\"umeler ayn\i\ k\"umelerin eleman\i d\i r:
  \begin{equation*}
    \Forall x\Forall y\Forall z(x=y\land x\in z\lto y\in z).
  \end{equation*}
\end{axiom}

\begin{theorem}\label{thm:sub}
  E\c sit k\"umeler ayn\i\ s\i n\i flar\i n eleman\i d\i r:
    her $\bm A$ s\i n\i f\i\ i\c cin
  \begin{equation*}
    \Forall x\Forall y(x=y\land x\in\bm A\lto y\in\bm A).
  \end{equation*}
\end{theorem}

\begin{proof}
  E\c sit $a$ ve $b$ k\"umeleri i\c cin,
  t\"umevar\i m ile
  her tek serbest de\u gi\c skeni $x$ olan
  $\phi$ form\"ul\"u i\c cin
  \begin{equation*}
    \phi(a)\liff\phi(b)
  \end{equation*}
  g\"osterece\u giz.
  Birincisi hari\c c,
  durumlar saf mant\i ktan do\u grudur.
  \begin{asparaenum}
    \item
  E\u ger bir $c$ k\"umesi i\c cin $\phi$
    \begin{compactenum}
    \item
      $c\in x$ ise,
      \Teoremde{thm:=}n iddia do\u grudur;
    \item
      $x\in c$ ise,
      E\c sitlik Aksiyomu'ndan iddia do\u grudur;
    \item
      $x\in x$ ise,
      $a\in a$ olsun.
      O zaman \Teoremde{thm:=}n $a\in b$,
      dolay\i s\i yla E\c sitlik Aksiyomu'ndan $b\in b$.
      Simetriden $a\in a\liff b\in b$.
    \end{compactenum}
  \item
    E\u ger $\phi$'nin bir $\psi$ form\"ul\"u oldu\u gu durumda
    iddia do\u gru ise,
    o zaman
     $\phi$'nin $\lnot\psi$ oldu\u gu durumda
    iddia do\u grudur.
  \item
    Benzer \c sekilde
    e\u ger $\phi$'nin bir $\psi$ veya bir $\chi$
    form\"ul\"u oldu\u gu durumda
    iddia do\u gru ise,
    $\phi$'nin $(\psi\land\chi)$
    oldu\u gu durumda
    iddia do\u grudur.
  \item
    E\u ger serbest de\u gi\c skenleri $x$ ve $y$ olan
    bir $\psi$ form\"ul\"u i\c cin,
    her $c$ k\"umesi i\c cin
    $\phi$'nin $\psi(x,c)$
    oldu\u gu durumda
    iddia do\u gru ise,
    o zaman
    $\phi$'nin $\Exists y\psi$ t\"umel evetlemesi
    oldu\u gu durumda
    iddia do\u grudur.\qedhere
  \end{asparaenum}
\end{proof}

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

\section{T\"umevar\i m}

\begin{theorem}\label{thm:cap-cup}
  \begin{align*}
  \bigcap\emptyset&=\universe,&\bigcup\emptyset&=\emptyset,
  \end{align*}
  ve ayr\i ca
  her $\bm A$ s\i n\i f\i\ i\c cin
  \begin{equation*}
    \Forall x
  \left(x\in\bm A\lto\bigcap\bm A\included x\included\bigcup\bm A\right),
  \end{equation*}
  ve ayr\i ca
  her $\bm B$ s\i n\i f\i\ i\c cin
  \begin{equation*}
  \bm A\included\bm B\lto\bigcap\bm B\included\bigcap\bm A
  \land\bigcup\bm A\included\bigcup\bm B.    
  \end{equation*}
\end{theorem}

\ktk

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

\begin{theorem}
  $\universe$ bir k\"ume de\u gildir,
  ama $\bm A$ bo\c s de\u gilse $\bigcap\bm A$ bir k\"umedir.
\end{theorem}

\ktk

\begin{axiom}[Bo\c s K\"ume]\label{ax:empty-set}
  $\emptyset$ s\i n\i f\i\
bir k\"umedir.
\end{axiom}

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

\begin{axiom}[Biti\c stirme]\label{ax:adjunction}
  Her $a$ ve her $b$ k\"umesi i\c cin $a\cup\{b\}$
%  \begin{equation*}
%    a\cup\{b\}
%  \end{equation*}
  s\i n\i f\i\
  bir k\"umedir.
\end{axiom}

\"Orne\u gin
tan\i ma g\"ore
\begin{equation*}
  (a,b)=\bigl\{\{a\},\{a,b\}\bigr\}
\end{equation*}
olabilir.

\begin{theorem}
  $\Forall x\Forall y\Forall z\Forall w
  \bigl((x,y)=(z,w)\liff x=z\land y=w\bigr)$.
\end{theorem}

\ktk

Ayr\i ca her k\"umenin ard\i l\i\ da bir k\"umedir.
Tan\i ma g\"ore
\begin{align*}
  1&=0',&
  2&=1',&
  3&=2',&
  &\dots
\end{align*}
olsun.
Bir $\bm A$ s\i n\i f\i\ i\c cin,
e\u ger
  \begin{compactenum}[1)]
  \item
    $0\in\bm A$,
  \item
    $\Forall x(x\in\bm A\lto x'\in\bm A)$
  \end{compactenum}
  ise, $\bm A$'ya \textbf{t\"umevar\i mli} densin.
\"Orne\u gin $\universe$ t\"umevar\i ml\i d\i r.
  
\begin{axiom}[Sonsuzluk]\label{ax:infinity}
T\"umevar\i ml\i\ bir k\"ume vard\i r.
\end{axiom}

Tan\i ma g\"ore $\upomega$,
t\"umevar\i ml\i\ k\"umelerin olu\c sturdu\u gu s\i n\i f\i n kesi\c simi
olsun.  B\"oylece
\label{omega}\label{X}\begin{equation*}
  \upomega=\bigcap\{X\colon 0\in X\land\Forall y(y\in X\lto y'\in X)\}.
\end{equation*}
Ayr\i ca
\begin{equation*}\label{N}
  \N=\upomega\setminus\{0\}
\end{equation*}
olsun.
Sonsuzluk ve Ay\i rma aksiyomlar\i\ sayesinde $\upomega$ ve $\N$,
k\"umedir.
Birincisi \textbf{do\u gal say\i lar\i} i\c cerir;
ikincisi,
\textbf{sayma say\i lar\i n\i.}
\begin{compactitem}
\item\label{k}
  $k$, $\ell$, $m$, ve $n$,
  sabit do\u gal say\i lar g\"osterecektir;
\item\label{i}
  $i$ ve $j$,
  do\u gal say\i\ de\u gi\c skendirler.
\end{compactitem}

\begin{theorem}[T\"umevar\i m]\label{thm:ind}\mbox{}
  \begin{compactenum}
    \item
      $\upomega$ t\"umevar\i ml\i d\i r.
      \item
  $\upomega$'n\i n tek t\"umevar\i ml\i\
        altk\"umesi kendisidir.
  \end{compactenum}
\end{theorem}

\begin{proof}
  \begin{asparaenum}
  \item
    $\upomega$ t\"umevar\i ml\i d\i r \c c\"unk\"u
    \begin{compactenum}
    \item
      her t\"umevar\i ml\i\ k\"ume $0$'\i\ i\c cerir,
      dolay\i s\i yla $0\in\upomega$;
    \item
      e\u ger $k\in\upomega$ ise,
      o zaman
      her \label{A-harfi}$A$ t\"umevar\i ml\i\ k\"umesi $k$'yi i\c cerir,
      dolay\i s\i yla $k'\in A$;
      sonu\c c olarak $k'\in\upomega$.
    \end{compactenum}
  \item
    E\u ger $A$ t\"umevar\i ml\i\ ise,
    o zaman
  \Teorem{thm:cap-cup} sayesinde
$\upomega\included A$.
    E\u ger ayr\i ca $A\included\upomega$ ise,
    o zaman $A=\upomega$.\qedhere
  \end{asparaenum}
\end{proof}

Sonraki teorem,
t\"umevar\i m\i n en basit a\c sik\^ar olmayan uygulamad\i r.

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

\ktk

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

\ktk

\"Ozaltk\"ume olma ba\u g\i nt\i s\i\
hem \textbf{yans\i mas\i zd\i r}
hem de
\textbf{ge\c ci\c slidir,}
\c c\"unk\"u s\i ras\i yla
\begin{gather*}
  \Forall x
  x\npincluded x,\\
  \Forall x\Forall y\Forall z
  (x\pincluded y\land y\pincluded z\lto x\pincluded z).
\end{gather*}
K\i saca \"ozaltk\"ume olma, bir \textbf{s\i ralamad\i r}.


\begin{theorem}\label{thm:n-trans}
  $\upomega$'da
  \begin{equation*}
    k\in n\lto k\pincluded n.
  \end{equation*}
\end{theorem}

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

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

\ktk

\textbf{Peano Aksiyomlar\i,}
a\c sa\u g\i daki \"u\c c sonu\c ctur:
\begin{compactenum}
  \item
  $0$'\i n ard\i l olmas\i
\item
  \Teorem{thm:ind}
\item
  \Teorem{thm:succ}
\end{compactenum}
Bunlardan do\u gal say\i lar\i n
matematikte kullan\i lan t\"um \"ozellikleri kan\i tlanabilir.
Matematikte do\u al say\i lar\i n\i n \emph{k\"ume} oldu\u gu
kullan\i lmaz,
ama bu \"ozellik ile
k\"umeler kuram\i nda do\u gal say\i lar\i n di\u ger \"ozelliklerini
kan\i tlamak daha kolayd\i r.

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

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

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

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

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

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

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

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

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

\ksk

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

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

\section{\"Ozyineleme}

Bir $\bm R$ ba\u g\i nt\i s\i\ i\c cin,
bir $\bm D$ s\i n\i f\i\ i\c cin,
e\u ger
\begin{equation*}
  \Forall x\Forall y\Forall z
  (x\in\bm D\land
  x\mathrel{\bm R}y
  \land
  x\mathrel{\bm R}z
  \lto
  y=z)
\end{equation*}
ise, o zaman $\bm D$'de $\bm R$ bir \textbf{g\"ondermeyi} tan\i mlar.
E\u ger bu g\"ondermeye \label{bm-F}$\bm F$ denirse,
o zaman $\bm D$,
$\bm F$'nin \textbf{tan\i m s\i n\i f\i d\i r,}
ve $x\in\bm D\land x\mathrel{\bm R}y$
form\"ul\"un\"un yerine
\begin{equation*}
  \bm F(x)=y
\end{equation*}
yaz\i labilir;
ayr\i ca $\bm F$'nin yerine
\begin{equation*}
  x\mapsto\bm F(x)
\end{equation*}
yaz\i labilir.
\"Orne\u gin $\universe$'de
  $x\mapsto x'$
g\"ondermesi vard\i r.

E\u ger bir $\bm E$ s\i n\i f\i\ i\c cin
\begin{equation*}
  \Forall x\Forall y(\bm F(x)=y\lto y\in\bm E)
\end{equation*}
ise, bu c\"umle
\begin{equation*}
  \bm F\colon\bm D\to\bm E
\end{equation*}
olarak yaz\i labilir.

$\Exists x\Exists y\bigl(z=(x,y)\land\bm F(x)=y\bigr)$
form\"ul\"u taraf\i ndan tan\i mlanm\i\c s s\i n\i f
\begin{equation*}
  \{(x,y)\colon\bm F(x)=y\}
\end{equation*}
olarak yaz\i labilir,
ve bu s\i n\i f,
$\bm F$'nin kendisi olarak anla\c s\i labilir.


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

Aksiyomda verilen k\"ume
\begin{align*}
  &\{\bm F(x)\colon x\in a\},&
  &\bm F[a]
\end{align*}
ifadelerinin biri ile g\"osterilebilir.
E\u ger $\bm F$'nin tan\i m s\i n\i f\i\ bir k\"ume ise,
o zaman Yerle\c stirme Aksiyomu'ndan
$\bm F$'nin kendisi de bir k\"umedir.


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


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

  \begin{proof}
    \.Istedi\u gimiz \"ozellikleri olan bir g\"onderme varsa,
    o zaman
    t\"umevar\i mdan
    tek bir \"ornek vard\i r.

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

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

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

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

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

  \"Ozyineleme y\"ontemiyle 
    $\upomega$'da toplama ve \c carpma i\c slemlerini tan\i mlayabiliriz:
    \begin{align*}
      k+(n+1)&=(k+n)+1,&k\cdot 1&=k,&k\cdot(n+1)&=kn+k.
    \end{align*}
    \c Simdi t\"umevar\i m ve kalan Peano Aksiyomlar\i\ ile
    toplaman\i n ve \c carpman\i n \"ozelliklerini kan\i tlayabiliriz;
    ayr\i ca $\upomega$'n\i n s\i ralamas\i n\i\ tan\i mlay\i p
    \"ozelliklerini kan\i tlayabiliriz.
    Ondan sonra
    $\N$,
    \label{Z}$\Z$,
    \label{Q}$\Q$ ve
    \label{R}$\R$ yap\i lar\i n\i\ elde edebiliriz.
  

    Do\u gal say\i lar, \emph{sonlu ordinallerdir};
    $\upomega$, sonsuz bir ordinald\i r.

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

\section{Ordinaller}

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

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

Ordinaller
\begin{equation*}\label{ON}
  \on
\end{equation*}
s\i n\i f\i n\i\ olu\c stururlar.
K\"u\c c\"uk Yunan harfleri
her zaman
$\on$'nin elemanlar\i\
g\"osterecektir.
\"Ozel olarak
\begin{compactitem}
  \item
\label{alpha}$\alpha$, $\beta$, $\gamma$ gibi harfler
sabit ordinaldirler,
ama
\item
  \label{xi}$\xi$, $\eta$, ve $\zeta$, ordinal de\u gi\c skendirler.
\end{compactitem}
\"Ornegin
\begin{equation*}
\{\xi\colon\phi(\xi)\}
=\{x\colon x\in\on\land\phi(x)\}.
\end{equation*}

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

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

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

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

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

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

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

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

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

\ktk

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

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

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

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

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

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

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

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

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

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

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

\ktk

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


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

\ktk

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

Varsa,
do\u grusal s\i ralanm\i\c s bir s\i n\i f\i n
bir $A$ altk\"umesinin
en k\"u\c c\"uk \"usts\i n\i r\i,
$A$'n\i n \textbf{supremumudur} ve
\label{sup}\begin{equation*}
  \sup A
\end{equation*}
olarak yaz\i l\i r.

\begin{theorem}\label{thm:sup}
  $\on$'n\i n her altk\"umesinin supremumu vard\i r.
  Asl\i nda
$B\pincluded\on$ ise
\begin{equation*}
  \sup B=\bigcup B.
\end{equation*}
\end{theorem}

\ktk


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



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


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

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

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

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


\begin{theorem}[Ordinal \"Ozyineleme]\label{thm:ord-rec}
Varsay\i mlar\i m\i z,
\begin{compactenum}[1)]
\item\label{theta}
$\theta\in\on$, 
\item
$\bm F\colon\on\to\on$.
\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 $\bm H(\beta')=\bm F\bigl(\bm H(\beta)\bigr)$,
\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 $h_{\alpha}(\beta')=\bm F\bigl(h_{\alpha}(\beta)\bigr)$,
\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\bigl(h_{\delta}(\delta)\bigr),&\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 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,
dolay\i s\i yla
\Teoreme{thm:sup} g\"ore
supremumu vard\i r.
Bu supremum,
\begin{equation*}
  	\sup_{\xi<\alpha}\bm F(\xi)
\end{equation*}
\c seklinde yaz\i labilir.

\begin{theorem}\label{thm:sup-a}
  Her $\alpha$ ordinali i\c cin
  \begin{equation*}
    \sup_{\xi<\alpha}\xi=
    \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
  \begin{equation}\label{eqn:sup}
    \bm F(\alpha)=\sup_{\xi<\alpha}\bm F(\xi)
  \end{equation}
\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
  \eqref{eqn:sup} do\u gru
\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_{\xi<\gamma}\bm F(\xi)&&\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\bigl(\sup(A)\bigr)=\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 $\alpha$ noktas\i ndaki s\"ureklili\u gi
ger\c cel analizdeki gibi tan\i mlan\i r.
\"Oyleyse e\u ger
\begin{equation*}
\epsilon_0<\bm F(\alpha)<\epsilon_1
\end{equation*}
ko\c sulunu sa\u glayan her $\epsilon_0$ ve $\epsilon_1$ i\c cin,
\begin{equation*}
  \delta_0<\alpha<\delta_1
\end{equation*}
ko\c sulunu sa\u glayan her $\delta_0$ ve $\delta_1$ i\c cin,
\begin{equation*}
\Forall{\xi}(\delta_0<\xi<\delta_1\lto\epsilon_0<\bm F(\xi)<\epsilon_1)
\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 $\epsilon_0=-1$ veya $\delta_0=-1$ olabilir.

\begin{lemma}
$\on$'de her i\c slem,
limit olmayan her noktada s\"ureklidir.
\end{lemma}

\klk

\begin{theorem}
  Kesin artan bir ordinal i\c slemin normal olmas\i\ i\c cin
  yeter ve gerek bir ko\c sul,
  i\c slemin s\"urekli olmas\i d\i r.
\end{theorem}

\ktk

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

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

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

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

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

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

\mbox{}
\end{proof}

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

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

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

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

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

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

\"Orne\u gin
\Sekle{fig:w+x} bak\i n.
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  \caption{$\eta=\upomega+\xi$ denkleminin grafi\u gi}\label{fig:w+x}  
\end{figure}
\Teoremi{thm:rec} kullanan resmi \"ozyineli tan\i ma g\"ore,
\begin{align*}
\upomega\cdot0&=0,&
	\upomega\cdot1&=\upomega,&
	\upomega\cdot(k+1)&=\upomega\cdot k+\upomega.
\end{align*}
Bu \c sekilde $\upomega\cdot n$, ``$\upomega$'d\i r $n$ kere''
veya ``$\upomega$'n\i n $n$ kat\i d\i r.''
Ayr\i ca
\begin{equation*}
\upomega^2=\upomega\cdot\upomega=\sup_{i<\upomega}(\upomega\cdot i).
\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]}\\
&=\bigl(\alpha+(\beta+\delta)\bigr)'&&\text{[\eqref{eqn:+'} tan\i m\i ndan]}\\
&=\bigl((\alpha+\beta)+\delta\bigr)'&&\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_{\xi<\delta}\bigl((\alpha+\beta)+\xi\bigr)&&\text{[\eqref{eqn:+lim} tan\i m\i]}\\
&=\sup_{\xi<\delta}\bigl(\alpha+(\beta+\xi)\bigr)&&\text{[\eqref{eqn:+assoc-lim} hipotezi]}\\
&=\alpha+\sup_{\xi<\delta}(\beta+\xi)&&\text{[$\xi\mapsto\alpha+\xi$ normaldir]}\\
&=\alpha+(\beta+\delta).&&\text{[\eqref{eqn:+lim} tan\i m\i]}\qedhere 
\end{align*}
  \end{asparaenum}
\end{proof}

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

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

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

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

\section{Hesaplamalar}

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

\begin{theorem}%\label{thm:k+w=w}
  $k<\upomega$ ise $k+\upomega=\upomega$.
(\Sekle{fig:x+w} bak\i n.)
\begin{figure}
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  \caption{$\eta=\xi+\upomega$ denkleminin grafi\u gi}\label{fig:x+w}  
\end{figure}
\end{theorem}

\begin{proof}
  $k+\upomega=\sup_{i<\upomega}(k+i)=\upomega$.
\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_{\alpha+\xi\leq\beta}\xi\\
	&=\sup_{\alpha+\xi\leq\beta}(\alpha+\xi)
	\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}\label{thm:w^2}
  E\u ger $\alpha<\upomega^2$ ise,
  o zaman bir ve tek bir \c sekilde
  \begin{equation}\label{eqn:a=wk+m}
    \alpha=\upomega\cdot k+m.
  \end{equation}
\end{theorem}

\begin{proof}
  $\upomega^2=\sup_{i<\upomega}\upomega\cdot i$ oldu\u gundan
  \begin{equation*}
    \{i\colon\alpha<\upomega\cdot i\}
  \end{equation*}
  k\"umesi bo\c s de\u gildir.
  Ayr\i ca k\"ume $0$'\i\ i\c cermez \c c\"unk\"u
  $\upomega\cdot0=0$.
  Sonu\c c olarak bir $k$ i\c cin
  \begin{gather*}
    k+1=\min\{i\colon\alpha<\upomega\cdot i\},\\
    \upomega\cdot k\leq\alpha<\upomega\cdot(k+1).
  \end{gather*}
  \Teorem{thm:subtraction} sayesinde bir $m$ i\c cin
  \eqref{eqn:a=wk+m} do\u grudur.
  \c Simdi baz\i\ $\ell$ ve $n$ i\c cin
  $\beta=\upomega\cdot \ell+n$ olsun.
  E\u ger $k<\ell$ ise, o zaman $\alpha<\beta$.
  E\u ger $k=\ell$ ama $m<n$ ise, o zaman $\alpha<\beta$.
\end{proof}

$\upomega^2$ k\"umesi 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}

\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_{\xi<\gamma}\alpha\cdot\xi.
\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_{i<\upomega}(\upomega^i).
\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

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

\ktk

\begin{theorem}\label{thm:w^k+w^m=w^m}
$k<m$ 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}
  $0<k$ ise $k\cdot\upomega=\upomega$.
(\Sekle{fig:xw} bak\i n.)
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  \centering
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Xi-axis
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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(10.24, 0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ticks on the Eta-axis
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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  \caption{$\eta=\xi\cdot\upomega$ denkleminin grafi\u gi}\label{fig:xw}  
\end{figure}

\end{theorem}

\ktk

\begin{lemma}\mbox{}\label{lem:w^w}
  \begin{compactenum}[(a)]
    \item
      E\u ger $0<\alpha<\upomega^{\upomega}$ ise,
      o zaman bir ve tek bir $k$ i\c cin
  \begin{equation*}
    \upomega^k\leq\alpha<\upomega^{k+1}.
  \end{equation*}
\item
  Bu durumda
      bir ve tek bir $m$ i\c cin
  \begin{equation*}
    \upomega^k\cdot m\leq\alpha<\upomega^k\cdot(m+1);
  \end{equation*}
  ayr\i ca $m>0$.
  \end{compactenum}
\end{lemma}

\klk[ \Teoremin{thm:w^2} kan\i t\i na bak\i n.]

\begin{theorem}\label{thm:w^2-CNF}
  E\u ger
  $0<\alpha<\upomega^{\upomega}$ ise,
  o zaman
  bir ve tek bir \c sekilde
  \begin{gather}\label{eqn:a-cnf}
    \alpha=\upomega^{k_0}\cdot m_0+\dots+\upomega^{k_n}\cdot m_n,\\\notag
    k_0>\dots>k_n,\\\notag
    \bigwedge_{i\leq n}m_i>0.
  \end{gather}
\end{theorem}

\ktk

\eqref{eqn:a-cnf} e\c sitli\u gindeki toplam,
$\alpha$'n\i n \textbf{Cantor normal bi\c cimidir.}
Ayr\i ca $0$'\i n Cantor normal bi\c cimi $0$'d\i r.

\begin{lemma}\label{lem:a+w^m=w^m}
$\alpha<\upomega^m$ ise $\alpha+\upomega^m=\upomega^m$.
\end{lemma}

\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{theorem}%\label{thm:mnk}
  $\alpha<\upomega^m$, $n>0$, ve $k>0$ ise
  \begin{equation*}
    (\upomega^m\cdot n+\alpha)\cdot k=\upomega^m\cdot n\cdot k+\alpha.
  \end{equation*}
\end{theorem}

\ktk

\begin{theorem}%\label{thm:mnk+}
  $\alpha<\upomega^m$, $n>0$, ve $k>0$ ise
  \begin{equation*}
    (\upomega^m\cdot n+\alpha)\cdot\upomega^k=\upomega^{m+k}.
  \end{equation*}
\end{theorem}

\ktk

\"Orne\u gin
\begin{gather*}
  (\upomega^5\cdot10+\upomega^3\cdot8+\upomega)\cdot6
=\upomega^5\cdot60+\upomega^3\cdot8+\upomega,\\
  (\upomega^5\cdot10+\upomega^3\cdot8+\upomega)\cdot\upomega
=\upomega^6,\\
  (\upomega^5\cdot10+\upomega^3\cdot8+\upomega)\cdot\upomega^{10}
=\upomega^{15}.
\end{gather*}
Ayr\i ca
\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}

\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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\uput[ur](! 0 10.24 1.28 sub){$\upomega^{\upomega^{\upomega}}$}
\uput[ur](0,10.24){$\upvarepsilon_0$}
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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{lemma}%\label{lem:a<w^(a+1)}
  Her $\alpha$ i\c cin $\alpha\leq\upomega^{\alpha}<\upomega^{\alpha+1}$.
\end{lemma}

\klk[ T\"umevar\i m kullan\i labilir.]


\begin{lemma}\mbox{}
  \begin{compactenum}
    \item
      E\u ger $0<\alpha$ ise,
      o zaman bir ve tek bir $\beta$ i\c cin
  \begin{equation*}
    \upomega^{\beta}\leq\alpha<\upomega^{\beta+1}.
  \end{equation*}
\item
  Bu durumda
      bir ve tek bir $m$ i\c cin
  \begin{equation*}
    \upomega^{\beta}\cdot m\leq\alpha<\upomega^{\beta}\cdot(m+1);
  \end{equation*}
  ayr\i ca $m>0$.
  \end{compactenum}
\end{lemma}

\klk[ Lemma \numaraya{lem:w^w} bak\i n.]

\begin{theorem}\label{thm:CNF}
  E\u ger $0<\alpha$ ise,
  o zaman
    bir ve tek bir \c sekilde
  \begin{gather}\label{eqn:a-cnb}
    \alpha=\upomega^{\beta_0}\cdot m_0+\dots+\upomega^{\beta_n}\cdot m_n,\\\notag
    \beta_0>\dots>\beta_n,\\\notag
    \bigwedge_{i\leq n}m_i>0.
  \end{gather}
\end{theorem}

\ktk

\eqref{eqn:a-cnb} e\c sitli\u gindeki
toplam,
$\alpha$'n\i n \textbf{Cantor normal bi\c cimidir.}

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

\klk[ (Lemma \numaraya{lem:a+w^m=w^m} bak\i n.)]

\begin{theorem}
$\alpha<\upomega^{\beta}$, $n>0$, ve $k>0$ ise
\begin{equation*}
  (\upomega^{\beta}\cdot n+\alpha)\cdot k
  =\upomega^{\beta}\cdot n\cdot k+\alpha.
\end{equation*}
\end{theorem}

\ksk

\begin{theorem}\label{thm:wbn+a}
$\alpha<\upomega^{\beta}$, $n>0$, ve $1\leq\gamma$ ise
\begin{equation*}
(\upomega^{\beta}\cdot n+\alpha)\cdot\upomega^{\gamma}=\upomega^{\beta+\gamma}.
\end{equation*}
\end{theorem}

\ktk

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

\begin{theorem}\label{thm:k^w}
  $0<k$ ve $\upomega\leq\alpha$ ise
  \begin{align*}
    k^{\upomega^{n+1}}&=\upomega^{\upomega^n},&
    k^{\upomega^{\alpha}}&=\upomega^{\upomega^{\alpha}}.
  \end{align*}
\end{theorem}

\ktk

\begin{lemma}
  Ard\i l olmayan her $\alpha$ i\c cin,
  bir $\beta$ i\c cin,
  \begin{equation*}
    \alpha=\upomega\cdot\beta.
  \end{equation*}
\end{lemma}

\klk

\begin{theorem}
  $\alpha<\upomega^{\beta}$, $n>0$, ve $\gamma$ limit ise
  \begin{equation*}
    (\upomega^{\beta}\cdot n+\alpha)^{\gamma}=\upomega^{\beta\cdot\gamma}.
  \end{equation*}
\end{theorem}

\ktk

\chapter{Kardinaller}\label{ch:card}


\section{E\c slemeler}

E\u ger $f\colon A\to B$ ise, \"u\c c \"ozel durum vard\i r.
\begin{compactenum}
  \item
E\u ger
  $\Forall{x_0}\Forall{x_1}(f(x_0)=f(x_1)\lto x_0=x_1)$ ise,
o zaman $f$ \textbf{birebirdir,}
bir \textbf{g\"ommedir,} ve
$A$'y\i\ $B$'ye \textbf{g\"om\"ur.}
Bu durumda
\begin{equation*}
\fbox{$A\preccurlyeq B$}
\end{equation*}
yaz\i l\i r.
\item
E\u ger $f[A]=B$ ise,
o zaman $f$, $B$'yi \textbf{\"orter.}
\item
E\u ger $f$ hem birebirdir hem de $B$'yi \"orter ise,
o zaman $A$ ve $B$ aras\i nda $f$ bir \textbf{e\c slemedir.}
Bu durumda $A$ ve $B$ \textbf{e\c sleniktir} ve
\begin{equation*}
  \fbox{$A\approx B$}
\end{equation*}
yaz\i l\i r.
\end{compactenum}

\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}

\section{Kardinaller}

E\u ger bir $A$ k\"umesi bir ordinal ile e\c slenik ise,
o zaman tan\i ma g\"ore
\begin{equation*}\label{card}
  \card A=\min\{\xi\colon\xi\approx A\}.
\end{equation*}
Bu ordinal, $A$'n\i n \textbf{kardinalidir.}
\c Simdi
\begin{equation*}\label{CN}
  \cn
  %=\{\text{kardinaller}\}
  =\cardinal[\on]
  =\{\xi\colon\Forall{\eta}(\xi\approx\eta\lto\xi\leq\eta)\}
\end{equation*}
olsun.


\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 $\alpha\approx\beta$
ve $\card{\alpha}\leq\card{\beta}$.
\end{proof}

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

\ktk

\label{kappa}$\kappa$ ve $\lambda$, her zaman kardinal olsun.

\section{Kardinal Toplama ve \c Carpma}

Tan\i ma g\"ore
\begin{align*}
  \kappa\oplus\lambda&=\card{\kappa+\lambda},&
  \kappa\otimes\lambda&=\card{\kappa\cdot\lambda}.
\end{align*}
Ordinal toplama ve \c carpma birle\c smeli oldu\u gundan
kardinal i\c slemler de birle\c smelidir.

Tan\i ma g\"ore
\begin{align*}
      A\times B
    &=\bigl\{z\colon\Exists x\Exists y
    \bigl(z=(x,y)\land x\in A\land y\in B\bigr)\bigr\}\\
    &=\{(x,y)\colon x\in A\land y\in B\}.
\end{align*}
Bu k\"ume, $A$ ve $B$'nin \textbf{Kartezyan \c carp\i m\i d\i r.}
\c Simdi
\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.}
\"Ozel olarak $A\sqcup A=A\times2$.

\begin{lemma}\label{lem:x.}
  Her $\alpha$ ve her $\beta$ i\c cin
  \begin{align*}
    \alpha\sqcup\beta&\approx\alpha+\beta,&
    \alpha\times\beta&\approx\alpha\cdot\beta.
  \end{align*}
\end{lemma}

\begin{proof}
  E\u ger
  \begin{align*}
    f(\xi,i)&=\alpha\cdot i+\xi,&
    g(\xi,\eta)&=\alpha\cdot\eta+\xi
  \end{align*}
  ise,
  o zaman
  \begin{align*}
    f&\colon\alpha\sqcup\beta\xrightarrow{\approx}\alpha+\beta,&
    f&\colon\alpha\times\beta\xrightarrow{\approx}\alpha\cdot\beta.\qedhere
  \end{align*}
\end{proof}

\begin{theorem}
  Her $\alpha$ ve her $\beta$ i\c cin
  \begin{align*}
    \alpha+\beta&\approx\beta+\alpha,&
    \alpha\cdot \beta&\approx\beta\cdot \alpha.
  \end{align*}
\end{theorem}

\ktk

\begin{corollary}
  $\cn$'de
  $\oplus$ ve $\otimes$ de\u gi\c smelidir.
\end{corollary}

\begin{lemma}\label{lem:wxw=w}
  $t_i=i\cdot(i+1)/2$ olmak \"uzere
  \begin{equation*}
    (i,j)\mapsto t_{i+j}+i\colon
    \upomega\times\upomega\xrightarrow{\approx}\upomega.
  \end{equation*}
\end{lemma}

\begin{lemma}
  Her sonsuz $\kappa$ i\c cin, bir $\alpha$ i\c cin,
  \begin{equation*}%\label{eqn:k=w^a}
    \kappa=\upomega^{\alpha}.
  \end{equation*}
\end{lemma}

\begin{proof}
  \Teoreme{thm:CNF} g\"ore bir $\beta$, bir $\gamma$,
  ve $\N$'de bir $n$ i\c cin
  \begin{align*}
    \gamma&<\upomega^{\beta},&
    \kappa&=\upomega^{\beta}\cdot n+\gamma.
  \end{align*}
  O zaman
  Lemma \numaraya{lem:x.} g\"ore
  $\kappa\approx\gamma+n\cdot\upomega^{\beta}$,
  dolay\i s\i yla
  \Teoreme{thm:wbn+a} g\"ore
  \begin{equation*}
    \kappa\approx\upomega^{\beta}.
  \end{equation*}
  E\u ger $\alpha=\min\{\xi\colon\upomega^{\xi}\approx\kappa\}$ ise,
  o zaman $\kappa=\upomega^{\alpha}$.
\end{proof}

\begin{lemma}
  E\u ger $\alpha>0$ ise,
  o zaman $\upomega^{\alpha}\times\upomega^{\alpha}\approx\upomega^{\alpha}$.  
\end{lemma}

\begin{proof}
  Lemma \numarada{lem:wxw=w}ki gibi $f(i,j)=t_{i+j}+i$ olsun.
  O zaman $f(0,0)=0$. Bir $n$ i\c cin, e\u ger
  \label{max}\begin{gather*}
    \alpha_0>\dots>\alpha_{n-1},\\
    \{b_0,\dots,b_{n-1}\}\pincluded\upomega,\\
    \{c_0,\dots,c_{n-1}\}\pincluded\upomega,\\
    0<\max(b_0,c_0)<\upomega\land\dots\land0<\max(b_{n-1},c_{n-1})<\upomega,\\
    \beta=\upomega^{\alpha_0}\cdot b_1+\dots+\upomega^{\alpha_{n-1}}\cdot b_{n-1},\\
    \gamma=\upomega^{\alpha_0}\cdot c_1+\dots+\upomega^{\alpha_{n-1}}\cdot c_{n-1}
  \end{gather*}
  ise
  \begin{equation*}
    \bm F(\beta,\gamma)
    =\upomega^{\alpha_0}\cdot f(b_0,c_0)+\dots
    +\upomega^{\alpha_{n-1}}\cdot f(b_{n-1},c_{n-1})
  \end{equation*}
  olsun.
  O zaman $\bm F$ birebirdir ve $\bm F[\upomega^{\alpha}\times\upomega^{\alpha}]
  =\upomega^{\alpha}$.
\end{proof}

\begin{theorem}\label{thm:cn.}
  E\u ger
  $2\leq\min(\kappa,\lambda)$ ve
  $\upomega\leq\max(\kappa,\lambda)$ ise, o zaman
  \begin{equation*}
     \kappa\oplus\lambda
    =\kappa\otimes\lambda
    =\max(\kappa,\lambda).
  \end{equation*}
\end{theorem}

\begin{proof}
  $\kappa\geq\lambda$ ise
  \begin{equation*}
    \kappa\leq\kappa\oplus\lambda
    \leq\kappa\oplus\kappa
    \leq\kappa\otimes2
    \leq\kappa\otimes\lambda
    \leq\kappa\otimes\kappa.\qedhere
  \end{equation*}
\end{proof}

Tan\i m k\"umesi $A$ olan,
de\u ger k\"umesi $B$'nin bir altk\"umesi olan g\"ondermeler
\begin{equation*}
  \fbox{${}^AB$}
\end{equation*}
k\"umesini olu\c stursun.

\begin{theorem}\label{thm:^A2}
${}^A2\approx\pow A$.
\end{theorem}

\ktk

E\u ger $f\in{}^AB$ ve $C\included A$ ise
\begin{equation*}
  f\restriction C=\bigl\{\bigl(x,f(x)\bigr)\colon x\in C\bigr\}
\end{equation*}
olsun.  O zaman $f\restriction C\in{}^CB$.

\begin{lemma}\label{lem:a^n}
  $0<n<\upomega\leq\alpha$ ise ${}^n\alpha\approx\alpha$.
\end{lemma}

\begin{proof}
  \Teoreme{thm:cn.} g\"ore bir $f$ i\c cin
  \begin{equation*}
    f\colon\alpha\times\alpha\xrightarrow{\approx}\alpha.
  \end{equation*}
  E\u ger $g_n\colon{}^n\alpha\xrightarrow{\approx}\alpha$ ise,
  o zaman
  \begin{equation*}
    x\mapsto f\bigl(g_n(x\restriction n),x(n)\bigr)
    \colon{}^{n+1}\alpha\xrightarrow{\approx}\alpha.\qedhere
  \end{equation*}
\end{proof}

\begin{lemma}\label{lem:uia}
  $\upomega\leq\alpha$ ise $\bigcup_{i\in\upomega}{}^i\alpha\approx\alpha$.
\end{lemma}

\begin{proof}
  Lemma \numaranin{lem:a^n} \emph{kan\i t\i ndan} bir $i\mapsto g_i$ i\c cin
  $\upomega$'da her $n$ i\c cin
  \begin{equation*}
    g_n\colon{}^n\alpha\xrightarrow{\preccurlyeq}\alpha.
  \end{equation*}
  Bir $f$ i\c cin e\u ger $a\colon n\to\alpha$ ise, o zaman
  $f(a)=\bigl(n,g_n(a)\bigr)$.  Bu \c sekilde
  \begin{equation*}
    f\colon\bigcup_{i\in\upomega}{}^i\alpha\xrightarrow{\preccurlyeq}
    \upomega\times\alpha.
    \qedhere
  \end{equation*}
\end{proof}

\c Simdi herhangi $A$ k\"umesi i\c cin
$n\in\upomega$ olmak \"uzere
\label{pow-2}\label{pown}\begin{equation*}
  \pow[n]A=\{X\in\pow A\colon\card X=n\}  
\end{equation*}
olsun, ve ondan sonra
\label{powf}\begin{equation*}
  \powf A=\bigcup_{i\in\upomega}\pow[i]A
\end{equation*}
olsun.

\begin{lemma}\label{lem:pwk}
  $\upomega\leq\kappa$ ise $\powf{\kappa}\approx\kappa$.
\end{lemma}

\begin{proof}
  \.Ilk olarak
  \begin{equation*}
    \xi\mapsto\{\xi\}\colon\kappa\xrightarrow{\approx}\pow[1]{\kappa}.
  \end{equation*}
  Bu e\c slemenin tersi $f_1$ olsun.
  Ayr\i ca
  $g\colon\kappa\times\kappa\xrightarrow{\approx}\kappa$ olsun.
  E\u ger
  \begin{equation*}
    f_m\colon\pow[m]{\kappa}\xrightarrow{\approx}\kappa
  \end{equation*}
  ise,
  o zaman
  \begin{equation*}
    (X,\xi)\mapsto g(f_m(X),\xi)\colon\pow[m]{\kappa}\times\kappa
    \xrightarrow{\approx}\kappa.
  \end{equation*}
  Ayr\i ca
  \begin{equation*}
    (X,\xi)\mapsto X\cup\{\max(X)+\xi\}\colon
    \pow[m]{\kappa}\times\kappa\xrightarrow{\approx}\pow[m+1]{\kappa},
  \end{equation*}
  dolay\i s\i yla \"oyle bir $f_{m+1}$ elde edilebilir ki
  \begin{equation*}
    f_{m+1}\colon\pow[m+1]{\kappa}\xrightarrow{\approx}\kappa.
  \end{equation*}
  Lemma \numaranin{lem:uia} kan\i t\i ndaki gibi
 $\powf{\kappa}\preccurlyeq\kappa$.
\end{proof}

\begin{theorem}
  $\alpha$ ve $\beta$ sonsuz ise
  \begin{equation*}
    \card{\beta^{\alpha}}
    =\max(\card{\alpha},\card{\beta}).
  \end{equation*}
\end{theorem}

\begin{proof}
  E\u ger $\gamma<\upomega^{\alpha}$ ve Cantor normal bi\c ciminde
  \begin{equation*}
    \gamma=\upomega^{\alpha_0}\cdot a_0+\dots+\upomega^{\alpha_{n-1}}\cdot a_{n-1}
  \end{equation*}
  ise,
  \begin{equation*}
    f(\gamma)=(i\mapsto a_i,\{\alpha_0,\dots,\alpha_{n-1}\}\}
  \end{equation*}
  olsun.  Bu \c sekilde
  \begin{equation*}
    f\colon\upomega^{\alpha}
    \xrightarrow{\approx}\bigcup_{i\in\upomega}{}^i\upomega\times\powf{\alpha},
  \end{equation*}
  dolay\i s\i yla Lemmalar \ref{lem:uia} ve \numarada{lem:pwk}n
  \begin{equation*}
    \card{\upomega^{\alpha}}=\card{\upomega}\otimes\card{\alpha}.
  \end{equation*}
  \Teoremde{thm:CNF},
  dolay\i s\i yla yukar\i da da,
   $\upomega$'n\i n yerine $\beta$ konulabilir.
\end{proof}

\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.

$\pow{\universe}=\universe$
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\preccurlyeq\pow A\land A\not\approx\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\"umesi]\label{ax:power-set}
Her $A$ k\"umesi i\c cin $\pow A$ s\i n\i f\i\ bir k\"umedir.
\end{axiom}

\c Su anda $A$'n\i n sonsuz oldu\u gu
$\pow A$ k\"umesinin kardinali var olup olmad\i\u g\i n\i\ bilmiyoruz.

\begin{theorem}
  $A\times B\included\pow{\pow{A\cup B}}$,
  dolay\i s\i yla k\"umedir.
\end{theorem}

\ktk

Tan\i ma g\"ore
\begin{equation*}
  \kappa^+=\{\xi\colon\xi\preccurlyeq\kappa\}
\end{equation*}
olsun.

\begin{theorem}[Hartogs]
  $\kappa^+\in\cn$, $\kappa^+>\kappa$, ve
  \begin{equation}\label{eqn:k+}
    \kappa^+
    =\min\{\xi\in\cn\colon\kappa<\xi\}.
  \end{equation}
\end{theorem}

\begin{proof}
  E\u ger $f\colon\alpha\xrightarrow{\preccurlyeq}\kappa$ ise,
  o zaman
\begin{equation*}
  \bigl\{\bigl(f(\xi),f(\eta)\bigr)\colon\xi\leq\eta<\alpha\bigr\}
  \in\pow{\kappa\times\kappa},
\end{equation*}
ve ayr\i ca elemandan $\alpha$ elde edilebilir.
Bu \c sekilde $\kappa^+$,
$\pow{\kappa\times\kappa}$'n\i n bir altk\"umesinin bir imgesidir,
dolay\i s\i yla $\kappa^+$ da bir k\"umedir.

\c Simdi $\kappa^+\included\on$ ve $\kappa^+$ ge\c ci\c sli
oldu\u gundan
  bir ordinaldir.
  Bu durumda $\kappa^+\notin\kappa^+$ oldu\u gundan $\kappa^+>\kappa$,
  ve tan\i mlardan \eqref{eqn:k+} \c c\i kar.
\end{proof}

$\kappa^+$, $\kappa$'n\i n \textbf{kardinal ard\i l\i d\i r.}
\"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 imgesi $\cn\setminus\upomega$.
%ve her sonsuz kardinal, bir $\alpha$ i\c cin, $\aleph_{\alpha}$'d\i r.
\end{theorem}

\ktk

\section{Se\c cme}

G\"ord\"u\u g\"um\"uz aksiyomlar, 
\textbf{Zermelo--Fraenkel} veya
\label{ZF}\textbf{ZF} aksiyomlar\i d\i r.
\c Simdi bir aksiyom daha kullanaca\u g\i z.

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

G\"odel'in kan\i tlad\i\u g\i\ bir teoreme g\"ore, 
{\zf} aksiyomlar\i n\i n bir \emph{modelinde,}
Se\c cim Aksiyomu (\eng{Axiom of Choice}) do\u grudur.
Cohen'in kan\i tlad\i\u g\i\ bir 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.

  Se\c cim Aksiyomu ile
  {\zf}, \label{ZFC}\textbf{ZFC} olur.
  \c Simdi her k\"umenin kardinali vard\i r.
  \"Orne\u gin
  tan\i ma g\"ore
  \begin{equation*}
    \lambda^{\kappa}=\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
Teoremler \ref{thm:k^w}, \ref{thm:^A2},
ve \numaraya{thm:pow} g\"ore
  \begin{align*}
    \card{2^{\upomega}}&=\aleph_0,&
      2^{\aleph_0}&>\aleph_0.
  \end{align*}
A\c sa\u g\i daki kurallar kolayd\i r.
\label{mu}\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}
2^{\lambda}<\kappa\lto
\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
$2^{\lambda}<\kappa$ ise
$\lambda\leq\kappa$, dolay\i s\i yla $\kappa$ sonsuzdur ve
\begin{equation*}
\kappa^{\lambda}
\leq(2^{\kappa})^{\lambda}
=2^{\kappa\cardprod\lambda}
=2^{\kappa}.\qedhere
\end{equation*}
\end{proof}
\"Orne\u gin
\begin{equation*}
  2\leq\kappa\leq2^{\aleph_0}<\mu\lto\kappa^{\aleph_0}=2^{\aleph_0}
  \leq \mu^{\aleph_0}\leq 2^{\mu}.
\end{equation*}

\c Simdi a\c sa\u g\i daki tan\i m yapar\i z.
\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 \label{CH}\textbf{KH,}
$\aleph_1=\beth_1$ \"onermesidir.
\textbf{Genelle\c stirilmi\c s Kontinuum Hipotezi} veya
\label{GCH}\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,
{\gch} 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{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<\kappa}\xi=\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_{i\in\upomega}\aleph_i.
  \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 $\{\aleph_i\colon i\in\upomega\}$, 
  $\aleph_{\upomega}$ 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,
  \label{cof}$\cof{\alpha}$
  \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.  
$\delta<\beta$ 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\bigl(k(\delta)\bigr),  
\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}

