Comprehensive Exam, Math 570, August 27, 2005.
Each of the five problems counts for 20 points. Explain your answers.
Conventions: The equality symbol = is considered as a logical symbol, Lis a
language, Nis the set of natural numbers (including 0), Zis the set of integers,
and Ris the set of real numbers. “Recursive” means the same as “computable.”
1. Let Lhave just a unary function symbol f.
(i) Find L-sentences σ1and σ2such that
6|=σ1,6|=σ2, σ16|=σ2,and σ2|=σ1.
(ii) Find a consistent L-sentence all whose models are infinite.
2. (i) Indicate (without proof) all automorphisms of (Z,0,+).
(ii) Is the element 1 definable in the structure (Z,0,+)? “Yes” means that there
exists a formula φ(x) in the language {0,+}such that for all integers k:
(Z,0,+) |=φ(k) if and only if k= 1.
(iii) Let Lconsist of the constant symbol 0, the binary relation symbol <and the
binary function symbol f. Indicate an L-formula ψ(x) with the following property:
for each function f:R2→Rthe formula ψ(x) defines in the structure (R,0, <, f)
the set of all a∈Rsuch that limy→+∞f(a, y) = 0.
3. Let Lconsist of one unary relation symbol U. Let Mbe an L-structure such
that both UMand M\UMare infinite.
(i) Let σnbe the L-sentence
[∃x1· · · ∃xn(^
i<j
xi6=xj∧^
i
U(xi))] ∧[∃x1· · · ∃xn(^
i<j
xi6=xj∧^
i
¬U(xi))].
(Note that σn∈Th(M).) Prove that {σn:n∈N} ` σfor all σ∈Th(M).
(ii) Prove that there is no θ∈Th(M) such that θ`σfor all σ∈Th(M).
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