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\def\gra{\mathop{\alpha}\nolimits}     
\def\grg{\mathop{\gamma}\nolimits}     
\def\grk{\mathop{\kappa}\nolimits}     
\def\gro{\mathop{\omega}\nolimits}     
\def\ssi{\mathop{\leftrightarrow}\nolimits}     
\def\all{\mathop{\forall}\nolimits}     
\def\proves{\mathop{\vdash}\nolimits}  
   


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\begin{center} {\bf Final Exam}\\
Math 610  \end{center}


Prove the statements below. Give complete proofs and write
legibly. You may use any non-human reference, but collaboration
with other humans is not permitted. If you have questions, you
may reach me at 233-4111 (home) or 631-8849 (office). Return the
test to me or to the secretary Patti by 5PM, Friday, May 9.



\begin{enumerate}
\item ZFC $\proves$ GCH $\:\ssi\: \all \grk \ge \gro\: (2^{< \grk} = \grk^{+}).$
\item Let $\grk = \aleph_{\gro}.$ Note that $\grk$ is a singular cardinal.
There is a function $f : \grk \to \grk$ such that \\
$\all \grg > 0 \: (f( \grg ) < \grg),$ but for all $\gra < \grk,$ $f^{-1}(\{ \gra \})$ is not stationary.
\item Let ${\bf M} \subseteq {\bf V} \models$ ZF be a transitive class which is a model of the Comprehension
Axiom Schema. Suppose furthermore that 
$${\bf V} \models \all x \subseteq {\bf M}\; \exists y \in {\bf M}\; (x \subseteq y).$$ 
Then ${\bf M}$ is a model of ZF.
\item Let $\grk > \gro$ be a regular cardinal and $f : \grk \to \grk$ a strictly increasing continuous
function. The set $\{ \gra : f ( \gra ) = \gra \}$ of fixed points of $f$ is a club in $\grk.$
\end{enumerate}

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