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\begin{document}
\bc
\bf
Mathematics 609 --- Final Exam  -- Version II
\ec
Answer all the following questions.  (Question 1
has changed from Version I.)  You may
use any source that you wish as long as it
isn't living.  The test is due at 3:00 PM
on Wednesday, December 17.  Please give it to
me personally or put it in my box on in the
mailroom on the third floor.

\be


\item Suppose that $X=\{\gf_i \st i\in \omega\}$
is a set of $\L$-sentences and $T$ a
theory such that for every model $\str A$ of
$T$, there is $\gf_i\in X$ such that
$\str A\models \gf_i$.  Show that
there is a finite subset $\{\gf_{i_1}, \dots
,\gf_{i_n}\}$ of formulas from $X$  such
that $T\proves \gf_{i_1}\lor \dots \lor
\gf_{i_n}$.



\item
Suppose that $\str A$ is a countably infinite $\L$-structure
in a countable language $\L$.
\be
\item
Show that $\str A$ is an atomic model of $T=\thy{\str A}$
if and only if for every finite $Y\subseteq A$, $\str A_Y$
is an atomic model of $T'=\thy{\str A_Y}$.
\item
With the same notation of the previous part, show that
$T$ is $\aleph_0$-categorical iff $T'$ is $\aleph_0$-categorical.
\ee


\item Show that if $\str A\els \str B$ and $\str B\els \str C$
then $\str A\els \str C$.

\item Suppose that $f$, $g$, $h$, and $k$ are primitive recursive
functions.  Let $p$ and $q$ be defined by simultaneous
recursion as follows.
\begin{eqnarray*}
p(x,0)&=&f(x)\\
q(x,,0)&=&g(x)\\
p(x,y+1)&=&h(x,y,p(x,y),q(x,y))\\
q(x,y+1)&=&k(x,y,p(x,y),q(x,y))
\end{eqnarray*}
Show that $p$ and $q$ are primitive recursive.

\item Show that $\mbox{\rm Tot}\equiv_1 \mbox{Inf}$.


\item Soare, page 32, 1.19.

\item Soare, page 32, 1.21.

\item \be\item
Soare, page 38, 3.8, part a. (Notation:  if $C$ and $D$
are sets, $C\oplus D$ denotes the set $\{2x\st x\in C\}\cup
\{2x+1\st x\in D\}$.

\item Use part a to give an alternate proof to Rice's Theorem,
namely that if  $A$ is an index set such that $A\ne \emptyset$
and $A\ne \omega$, then $A$ is not computable.
\ee

\item Soare, page 40, 3.16.

\item Soare, page 43, 4.10.
(The notation $\lambda x[g(x,y)]$ is notation for the function
$h(x)=g(x,y)$; in other words, the function of $x$ with $y$
fixed.)


\ee
\end{document}

