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\noindent\begin{minipage}[b]{0.25\textwidth}
Bu resimde von Neumann'\i n tan\i m\i na g\"ore ilk alt\i\ do\u gal
say\i\ g\"or\"ulebilir.   Burada 0, bo\c
s k\"umedir.  Resimdeki her \c cember, bo\c s olmayan bir k\"umedir, ve
bu k\"umenin elemanlar\i, \c cemberin i\c cindekilerdir.
Ayn\i\ elemanlar\i\ olan k\"umeler, birbiriyle ayn\i d\i r.  Dolay\i
s\i yla resimde 32 tane s\i f\i r ve 31 tane \c cember oldu\u gu
halde, asl\i nda sadece alt\i\ tane farkl\i\ k\"ume vard\i r.  Bu
k\"umeler, ilk alt\i\ do\u gal say\i d\i r, yani 0, 1, 2, 3, 4, ve 5.
Ayr\i ca
\begin{gather*}
5=\{0,1,2,3,4\},\\
4=\{0,1,2,3\},\\
3=\{0,1,2\},\\
2=\{0,1\},\\
1=\{0\}.
\end{gather*}
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\begin{minipage}[b]{0.25\textwidth}
Bir k\"umenin bir eleman\i n\i n elemanlar\i, o k\"umenin
eleman\i\ olmayabilir.  
Asl\i nda \emph{Temellendirme Aksiyomuna} g\"ore her bo\c s olmayan
$a$ k\"umesinin \"oyle bir $b$ eleman\i\ vard\i r ki $a$ ile $b$,
birbirinden ayr\i kt\i r, yani $a\cap b=0$.

E\u ger bir k\"umenin her eleman\i n\i n
elemanlar\i, o k\"umenin eleman\i ysa, bu k\"ume \emph{ge\c ci\c
  slidir.}  Von Neumann do\u gal say\i lar\i, ge\c ci\c slidir.  Ayr\i
ca bir von Neumann do\u gal say\i s\i n\i n t\"um elemanlar\i, ge\c
ci\c slidir.  Asl\i nda tan\i ma g\"ore elemanlar\i\ ge\c ci\c sli
olan ve kendisi ge\c ci\c sli olan bir k\"ume, bir \emph{ordinaldir.}
E\u ger $\alpha$ bir ordinal ise, o zaman $\alpha\cup\{\alpha\}$
k\"umesi de, bir ordinaldir, ve bu ordinal, $\alpha$ ordinalinin
\emph{ard\i l\i d\i r.}  E\u ger bir ordinal, ya bo\c s ya ard\i l,
ve dahas\i\ her eleman\i,  ya bo\c s ya ard\i l ise, o zaman bu
ordinal, bir \emph{do\u gal say\i d\i r.}
\end{minipage}
\\
\vfill
\centering
{\VERYHuge von Neumann do\u gal say\i lar\i}\\
\mbox{}\\
{\footnotesize David Pierce\hfill
\url{dpierce@msgsu.edu.tr}\hfill
\url{http://mat.msgsu.edu.tr/~dpierce/}
\hfill Matematik B\"ol\"um\"u, Mimar Sinan G\"uzel Sanatlar
\"Universitesi}
\end{document}

