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\title{Philosophy 152: Computability \& Logic}
\author{Lucas Garron}
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\section*{Mar 28, 2011}

\li
\item Frege: \emph{Grundlagen der Arithmetic}, 1880s
\item Cantor: Infinite set theory, 1880s.
\item Russell's Paradox: $S = \{x: x \not\in x\}$.
\item Russell and Whitehead: \emph{Principia Mathematica}.
\item Zermelo and ZF set theory.
\item Goal of foundations of mathematics: A consistent, formal system that proves all true statements of mathematics.
\item Hilbert: Hilbert's Program. (including: a finitary proof of consistency)
\lid

{\theorem (G\"{o}del's First Incompleteness Theorem)} For any sufficiently strong (Peano?) consistent formal system, there is a true statement that is not provable.

{\theorem (G\"{o}del's Second Incompleteness Theorem)} No sufficiently strong consistent formal system can prove its own consistency.

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