LOGIC COMPREHENSIVE EXAM (MATH 570),
AUGUST 2006
There are 5 problems. Each problem is worth 20 points, for a total of 100 points. To
receive credit, each of your solutions must be justified.
Convention. In the exercises Lwill be a language and =is considered a logical symbol.
Any model-theoretic structure is by convention non-empty.
1. EXERCISE
Let for each n∈Na function fn:N→Nbe given. Show that there is a function
g:N→Nsuch that for each nand all k≥n,fn(k)< g(k).
2. EXERCISE
Let Lbe the first order language whose non-logical symbols consist of a constant sym-
bol 0for the number zero, a unary function symbol Sfor the successor function, a binary
predicate symbol <for the ordering relation, and two binary function symbols +and ×
for addition and multiplication respectively. Let Σbe any set of sentences in Lsuch that
Σ`σwhenever σis a quantifier free sentence in Lthat is true in the standard model of
arithmetic (N,0, S, <, +,×). Let also Σcontain the sentence
∀x∀y∀z(x+y=x+z→y=z).
(1) Let f:Nk→Nbe a function. Define what it means for fto be representable (as
a function) in Σ.
(2) Define f:N2→Nby f(n, m) = max{0, n −m}. Show that fis representable
in Σ.
3. EXERCISE
Let Lbe a language with at least one constant symbol. Let φ(x)be a quantifier free
formula. Show that
` ∃x φ
if and only if there exist variable free terms t1, . . . , tnsuch that
`φ(t1)∨ · ·· ∨ φ(tn).
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