\documentclass[12pt]{amsart}
\usepackage{amsmath,amsthm,amsfonts,latexsym,amssymb,amscd}
\pagestyle{headings}
\setlength{\textwidth}{468true pt}
\setlength{\headheight}{0true pt} % needed for 12pt documents
\setlength{\oddsidemargin}{0 truept}
\setlength{\evensidemargin}{0 truept}
\setlength{\textheight}{580true pt}
\newtheorem{thm}{Theorem} %[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lm}[thm]{Lemma}
\newtheorem{prp}[thm]{Proposition}
\newtheorem{clm}[thm]{Claim}
%
\theoremstyle{definition}
\newtheorem{df}[thm]{Definition}
\newtheorem{exmp}[thm]{Example}
\newtheorem{exmps}[thm]{Examples}
\newtheorem{prb}[thm]{Problem}
%
%\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
%
\numberwithin{equation}{section}
%
\newcommand{\thref}[1]{Theorem~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lmref}[1]{Lemma~\ref{#1}}
%
\DeclareMathOperator{\satf}{\models}
\DeclareMathOperator{\miff}{\Longleftrightarrow} 
                                           % math `if and only if'
\DeclareMathOperator{\mimplies}{\Longrightarrow} 
                                           % math `implies'
\DeclareMathOperator{\mimplied}{\Longleftarrow}  
                                           % math `implied by'
\newcommand{\set}[2]{\{#1\,:\,\text{#2}\}} % { x : ... }
\DeclareMathOperator{\id}{\textsl{id}}     % The identity map
\DeclareMathOperator{\miffdef}{\stackrel{def}{\miff}}
                                           % defining iff
\DeclareMathOperator{\eqdef}{\stackrel{def}{=}}
                                           % defining equation

\DeclareMathOperator{\neareq}{\stackrel{\ae}{=}}
\DeclareMathOperator{\aeqv}{\stackrel{ae}{\equiv}}
\DeclareMathOperator{\aeq}{\stackrel{ae}{=}}
%
\DeclareMathOperator{\BZ}{\mathbb{Z}} % integers
\DeclareMathOperator{\BQ}{\mathbb{Q}} % rationals
\DeclareMathOperator{\BR}{\mathbb{R}} % reals
\DeclareMathOperator{\BC}{\mathbb{C}} % complex numbers
\DeclareMathOperator{\BD}{\mathbb{D}} % 
\DeclareMathOperator{\BE}{\mathbb{E}} % 
\DeclareMathOperator{\BF}{\mathbb{F}} % 
\DeclareMathOperator{\BK}{\mathbb{K}} % 
\DeclareMathOperator{\BL}{\mathbb{L}} % 
\DeclareMathOperator{\re}{Re}         % real part
\DeclareMathOperator{\im}{Im}         % imaginary part
\DeclareMathOperator{\Ker}{Ker}
%
\DeclareMathOperator{\meet}{\wedge}
\DeclareMathOperator{\join}{\vee}
\DeclareMathOperator{\union}{\cup}
\DeclareMathOperator{\inter}{\cap}  % intersection
\DeclareMathOperator{\iso}{\cong}   % isomorphic
\DeclareMathOperator{\piso}{\simeq} % polynomially isomorphic
\newcommand{\Star}{^ *}
\DeclareMathOperator{\compose}{{\scriptstyle \circ}} % f o g
%
\newcommand{\wec}[1]{{\mathbf{#1}}}  % notation for vectors in algebras
%\newcommand{\wec}[1]{{\bar{#1}}} % which one do you prefer?
\newcommand{\wrel}[1]{\;#1\;}     % a\wrel{\alpha}b = a \alpha b
%              % this provides a flexible way to change the spacing
%
\newcommand{\m}[1]{{\mathbf{\uppercase{#1}}}}
\newcommand{\vr}[1]{{\mathcal{\uppercase {#1}}}}
%
\DeclareMathOperator{\Var}{\mathbf{V}}
\DeclareMathOperator{\Ho}{\mathbf{H}}
\DeclareMathOperator{\Su}{\mathbf{S}}
\DeclareMathOperator{\Pd}{\mathbf{P}}
\DeclareMathOperator{\Iso}{\mathbf{I}}
\DeclareMathOperator{\Ps}{\mathbf{P}_s}
\newcommand{\fin}[1]{{\mathbf{#1}}_{\text{\upshape fin}}}
%
\DeclareMathOperator{\Pol}{Pol}
\DeclareMathOperator{\Clo}{Clo}
\DeclareMathOperator{\Con}{Con}
\DeclareMathOperator{\Conb}{\mathbf{Con}} % Con(A) as a lattice
\DeclareMathOperator{\cg}{Cg}
\DeclareMathOperator{\sg}{Sg}
\newcommand{\Cg}[1]{{\cg}^{\m{#1}}}
\newcommand{\Sg}[1]{{\sg}^{\m{#1}}}
%
\newcommand{\M}[3]{\text{\upshape M}_{\m{#1}} (#2, #3)}
\newcommand{\minl}[1]{\text{\upshape Min}_{\m{#1}}}
%
\DeclareMathOperator{\typ}{typ}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\Sep}{Sep}
\DeclareMathOperator{\ssb}{ssb}
\DeclareMathOperator{\sst}{sst}
%
\DeclareMathOperator{\utyp}{\mathbf{1}}% unary      
\DeclareMathOperator{\atyp}{\mathbf{2}}% affine     
\DeclareMathOperator{\btyp}{\mathbf{3}}% boolean    
\DeclareMathOperator{\ltyp}{\mathbf{4}}% lattice    
\DeclareMathOperator{\styp}{\mathbf{5}}% semilattice
\DeclareMathOperator{\ityp}{\mathbf{i}}
\DeclareMathOperator{\jtyp}{\mathbf{j}}
%
\DeclareMathOperator{\solv}{\stackrel{s}{\sim}}
\DeclareMathOperator{\ssolv}{\stackrel{ss}{\sim}}
%
\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}
%
\newcommand{\lb}{\langle} % <...>
\newcommand{\rb}{\rangle}
\newcommand{\lv}{{[\![}}  % [[...]]
\newcommand{\rv}{{]\!]}}
%
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
%
\newcommand{\vep}{\varepsilon}
\newcommand{\di}{\diamond}
%
\DeclareMathOperator{\Aut}{{\rm Aut}}
\DeclareMathOperator{\Sub}{{\rm Sub}}

\begin{document}


\noindent
{\large \bf The Axioms of Set Theory.}

\bigskip

\noindent
{\bf Equality}

\medskip

\begin{enumerate}
\item[(1)] (Extensionality) Two sets are equal if they have the 
same elements.
\end{enumerate}


\bigskip

\noindent
{\bf Existence of Special Sets}

\medskip

\begin{enumerate}
\item[(2)] (Empty Set) There is a set with no elements.

%\noindent
%(Call it the empty set and denote it by $\emptyset, \{\;\}$ or $0$.)
\smallskip
\item[(3)] (Infinity) There is an inductive set.
%\noindent
%(Equivalently, there is a set containing
%$\mathbb N=\{0,1,2,\ldots\}$.)
\end{enumerate}


\bigskip

\noindent
{\bf Creation of New Sets}

\medskip

\begin{enumerate}
\item[(4)] (Pairing) Given sets $A$ and $B$,
  there is a set whose only elements are $A$ and $B$.
(I.e., the unordered pair $\{A,B\}$ is a set.)
\smallskip
\item[(5)] (Union)
Given a set $A$, the collection of
  all elements of elements of $A$ is a set.
  It is denoted $\bigcup A$ and called the union of $A$.
%  If $I$ is a set, and $A_i$ is a set
%for each $i\in I$, then $\bigcup_{i\in I} A_i$ is a set.
\smallskip
\item[(6)] (Power Set) Given a set $A$, 
  the collection of subsets of $A$ is a set.
  It is denoted ${\mathcal P}(A)$ and called the
  power set of $A$.
%  ${\mathcal P}(A)$ is a set. (${\mathcal P}(A)$ -- the power set of $A$ --
%  is the collection of all subsets of $A$.)
\smallskip
\item[(7)] (Separation) Given a set $A$ and 
  a property $P$ given by a formula, there is a set
  whose elements are exactly those elements of $A$
  that satisfy $P$.
($\{x\in A\;|\;P(x)\}$ is a set.)
\smallskip
\item[(8)] (Replacement)
Given a set $A$ and a function $F$ given by a formula, 
then \\
$\{F(x)\;|\;x\in A\}$ is a set.
\smallskip
\item[(9)] (Choice) 
  Given $A=\{X_i\;|\;i\in I\}$, a set
  of nonempty pairwise-disjoint
sets, there is a set $C$ that intersects each $X_i$
in exactly one element.
%such that 
%$|A_i\cap C|=1$ for every $i$.
\end{enumerate}


\bigskip

\noindent
{\bf Sets have Special Properties}

\medskip

\begin{enumerate}
\item[(10)] (Foundation)
If $A$ is a nonempty set, then there is an $x\in A$ 
such that $x$ and $A$ are disjoint.
$x$ is called an $\in$-minimal element of $A$.
%There is no infinite sequence of sets $\cdots A_3\in A_2\in A_1\in A_0$.
\end{enumerate}


\end{document}

