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\begin{document}
\bc
\bf
Mathematics 609 --- Midterm Test
\ec
Answer all the following questions.  You may
use any source that you wish as long as it
isn't living.  The test is due at 5:00 PM
on Wednesday, October 29.
\be
\item
Given a set of students and a set of
classes, suppose that each student
wants to take a finite number of classes
and that each class has a finite enrollment
limit.  Suppose that each finite set
of students can be accomodated.  Show
that they all can be accomodated.
(This problem can be done using only
results from propositional logic.  Let
$P_{i,j}$ be a propositional variable
representing the fact that student $i$
wants course $j$.)


\item Problem III.8.1

\item Problem IV.1.2



\item
Let $T$ be a theory with an infinite
set $\Gamma$ of axioms.  Suppose that
for every finite set $\Delta\subseteq \Gamma$,
there is a sentence $\phi\in T$ such that
$\Delta\not\proves\phi$.  (That is, no
finite subset of $\Gamma$ is a set of axioms
for $T$.)  Show that there is no finite
set of axioms for $T$.

\item
Let $T$ be true arithmetic; that is
let $T=\thy{\str N}$.  In this problem
you will show that $T$ has uncountably
many nonisomorphic countable models.

\be
\item Given a prime number $p$, write down a
formula
$\gf_p(v)$  in the language of arithmetic
that
defines the relation ``v is divisible by the
prime number p.''
\item Let $S$ be any set of prime numbers.
Show that there is countable model $\str A$ of
$T$ and $a\in A$ such that if $p\in S$ then $\str
A\models\gf_p[a]$ and if $p\not\in S$ then
$\str A\models\neg\gf_p[a]$.
\item Show that there are uncountably many
countable models of $T$.
(Note that the number of sets $S$ of prime
numbers is uncountable. So, for example,
a single countable model $\str A$ will not
satisfy the property of part $b$ for every
$S$.)
\ee
\item Suppose that $T$ is a consistent
theory in a countable language such that
$n(T,\aleph_0)=\aleph_0$.  Show that there
is a sentence $\phi$ such that $T\cup \phi$
is complete.
\item Let $\L=\{0,S\}$ be the language
with a constant symbol $0$ and a unary
function symbol $S$.
Consider the following set of axioms.
\begin{itemize}
\item $\forall x \forall y (S(x)=S(y)\implies
x=y)$
\item $\forall x (x\ne 0\biimp  \exists y (
y=S(x)))$
\item for each $n$, the sentence
$\forall x (S^n(x)\ne x)$ where $S^n(x)=S(S(\dots
S(x)\dots ))$
\end{itemize}
Note that the natural numbers with the
usual successor function and 0 satisfy
these axioms.
\be
\item Show that $T$ is not finitely
axiomatizable.
\item Show that $T$ is not $\aleph_0$-categorical.
\item Show that $T$ is $\kappa$-categorical
for all $\kappa>\aleph_0$.
\ee
\item Suppose that $\str A_i$ are structures
such that $\str A_i\els \str A_{i+1}$ for
every $i\in \N$.
 Show that there is a natural way to
define a structure $\str A$ so that the
universe of $\str A$ is the union of the
sets $A_i$ and that for this
structure $\str A$ we have $\str A_i\els \str A$
for all $i$.
\item Show that the theory of dense linear
orderings {\bf with} endpoints is complete.
\item Suppose that $\{\gf_i \st i\in \N\}$
is a set of $\L$-sentences and $T$ a
theory such that for each $i$ there is
a model $\str A_i$ of $T$ such that
$\str A_i\models \gf_i$.  Show that
there is a finite subset $\{\gf_{i_1}, \dots
,\gf_{i_n}\}$ of these formulas such
that $T\proves \gf_{i_1}\lor \dots \lor
\gf_{i_n}$.


\ee
\end{document}

