Computability and Logic

PROBLEMS 151
12.12 Show that if two models of Eq are isomorphic, then the equivalence relations
of the models have the same signature.
12.13 Suppose E 1 and E 2 are equivalence relations on denumerable sets X 1 and
X 2 both having the signature σ (n) = 0 for n ≥ 1 and σ (0) =∞, that is, both
having infinitely many equivalence classes, all infinite. Show that the models
involved are isomorphic.
12.14 Show that two denumerable models of Eq are isomorphic if and only if they
have the same signature.
In the remaining problems you may, when relevant, use the Löwenheim–Skolem
and compactness theorems, even though the proofs have been deferred to the
next chapter.
12.15 Show that:
(a) Ɣ is unsatisfiable if and only if ∼C 1 ∨···∨∼C m is valid for some
C 1 , ..., C m in Ɣ.
(b) D is a consequence of Ɣ if and only if D is a consequence of some finite
subset of Ɣ.
(c) D is a consequence of Ɣ if and only if ∼C 1 ∨···∨∼C m ∨ D is valid
for some C 1 , ..., C m in Ɣ.
12.16 For any prime p = 2, 3, 5, ..., let D p (x) be the formula ∃y p · y = x of the
language of arithmetic, so that for any natural number n, D p (n) is true if
and only if p divides n without remainder. Let S be any set of primes. Say
that a nonstandard model M of arithmetic encrypts S if there is an individual
m in the domain |M| such that M |= D p [m] for all p belonging
to S, and M |= ∼ D p [m] for all p not belonging to S. Show that for any
set S of primes there is a denumerable nonstandard model of arithmetic that
encrypts S.
12.17 Show that there are nonenumerably many isomorphism types of denumerable
nonstandard models of arithmetic.
12.18 Show that if two sentences have the same enumerable models, then they are
logically equivalent.
12.19 Work with a language whose only nonlogical symbol is a single two-place
predicate