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\textbf{Final Exam., Math. 609, Fall, 2002}
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You will have 24 hours to work on this exam. You may use your notes. If
the statement of some problem is unclear, please ask me about
it.
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1. Let $T$ be an binary branching tree, consisting of finite
sequences of $0$'s and $1$'s, closed under initial segments. A path
through $T$ is a function $\pi\in 2^{\omega}$ such that for all $n$,
$\pi|n$ is in $T$. Consider the propositional language $S =
\{P_n:n\in\omega\}$. For each $\sigma\in T$, let $\varphi_\sigma$ be the
conjunction of those $P_n$ such that $\sigma(n) = 1$ and the negation of
those $P_n$ such that $\sigma(n) = 0$. For each $k$, there are only
finitely many sequences $\sigma$ of length $k$ in $T$. Let $\psi_k$ be
the disjunction of $\varphi_\sigma$ over these $\sigma$. Let $\Gamma =
\{\psi_k:k\in\omega\}$. Show that if $T$ is infinite, then $\Gamma$ is
consistent. Conclude that $T$ has a path.
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2. Every Archimedean ordered field can be embedded in the ordered
field of real numbers. Using this fact, show that if $T$ is the set of
elementary first order sentences true in all Archimedean ordered
fields, then $T$ has models which are not Archimedean.
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3. Let $T$ be the set of all sentences true in $(\omega,<)$. Show that
$T$ has a model with a subset having the order type of the rationals.
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4. For a countable, complete elementary first order theory $T$,
let $I(T,\aleph_0)$ be the number of isomorphism types of
countable models of $T$.
(a) Give an example such that $I(T,\aleph_0) = 1$.
(b) Give an example such that $I(T,\aleph_0) = 4$.
(c) Give an example such that $I(T,\aleph_0) = 2^{\aleph_0}$.
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[You do not need to give proofs---just describe the examples.]
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5. Let $T$ be the theory of non-trivial vector spaces over the
rationals.
(a) Describe (in terms of dimension) the prime model of $T$.
(b) Describe (again in terms of dimension) the countable saturated model
of~$T$.
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6. Recall that the Kleene $T$-predicate is
primitive recursive---$T(e,x,c)$ iff $c$ is a halting computation for
$\varphi_e(x)$. Using this, explain why, for $a$ and $e$ such that
$a\in W_e$, $W_e$ is the range of a primitive recursive function.
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7. Let $I = \{e:W_e\not=\emptyset\}$.
(a) Show that $I$ is c.e.
(b ) Show that $I$ is not computable.
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8. Show that there is a non-computable set $A$ whose jump, $A'$
is computable relative to $K$. You may wish to use the following
outline.
$R_2e$: $\varphi_e\not=\chi_A$.
$R_2e+1$: Put $e$ into $A'$, if possible.
The construction proceeds in stages. At stage $s+1$,
you have determined a finite initial segment $\sigma_s$ of
$\chi_A$, such that $\sigma_s$ has length at least $s$, and guarantees
satisfaction of $R_k$, for all $k < s$, and you want to define
$\sigma_{s+1}$ to guarantee satisfaction of $R_s$. The sequence
$(\sigma_s)_{s\in\omega}$ should be computable relative to $K$.
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