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\title{Foundations of Mathematics}
\author{Last Updated: June 1996}
\date{}

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\maketitle

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\section{First-order logic}

\begin{enumerate}
\item Propositional logic: provability, truth tables, consistency,
  compactness, completeness.
\item First-order predicate logic: syntax and semantics.
  \begin{itemize}
  \item Deduction systems and formal proofs.
  \item Consistency, completeness and decidability of theories: the
    methods of elimination of quantifiers and Vaught's Test.
  \item Godel's Completeness Theorem.  The Henkin proof.  The
    Compactness Theorem and its applications.
  \item Elementary Model Theory.  Elementary substructures and the
    Lowenheim-Skolem Theorem.
  \item Godel's Incompleteness Theorem.  Applications to undecidable
    theories.
  \end{itemize}
\end{enumerate}

\section{Set Theory}

\begin{enumerate}
\item Axiomatic set theory.  The systems ZF and ZFC.  Relations
  between sets and classes.
\item Principles of transfinite induction and recursion, and
  applications.
\item The definitions of ordinal and cardinal numbers.  Cardinal and
  ordinal arithmetic with and without the Generalized Continuum
  Hypothesis.
\item Natural models of set theory and parts thereof.  Reflection
  principles.
\item Transfinite trees, closed unbounded and stationary sets.
\end{enumerate}

\section{Recursive Function Theory}

\begin{enumerate}
\item Definition of recursive and partial recursive functions.
  Recursive and recursively enumerable sets.  Definability in
  arithmetic of recursive and r.e. sets.  Church's Thesis.
\item   Unsolvability of the halting problem, and sample
applications. 
\item Elementary recursive function theory.  The recursion theorem and
  the enumeration and parametrization ($s$-$m$-$n$) theorems.
\item Turing reducibility and degrees of unsolvability. The
  arithmetical hierarchy, $\Sigma_n$ and $\Pi_n$-sets for each
  integer~$n$.
\end{enumerate}

\section*{References}

\subsubsection*{First-order logic}

\begin{enumerate}
\item Enderton, \textit{A Mathematical Introduction to Logic}
\item Chang and Keisler, \textit{Model Theory}
\end{enumerate}

\subsubsection*{Set Theory}

\begin{enumerate}
\item Kunen, \textit{Set Theory}
\item Jech, \textit{Set Theory}
\item Roitman, \textit{Introduction to Modern Set Theory}
\end{enumerate}

\subsubsection*{Recursion Theory}

\begin{enumerate}
\item Odifreddi, \textit{Classical Recursion Theory}
\item Cutland, \textit{An Introduction to Recursive Function Theory}
\item Rogers, \textit{Theory of Recursive Functions and Effective
    Computability}
\end{enumerate}

The books listed above are only the most frequently recommended texts.
There are many others that may be quite good.

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