Computability and Logic

PROBLEMS 231
every axiom of Q is a theorem of T , then the set of sentences of the language of
arithmetic whose translations are theorems of T is a consistent, axiomatizable
extension of Q.
17.10 Show that under the hypotheses of the preceding problem, T is incomplete
and undecidable.
17.11 Let L be a language, N(u) a formula of L. For any sentence F of L, let the
relativization F N be the result of replacing each universal quantifier ∀x in F
by ∀x(N(x) →···) and each existential quantifier ∃x by ∃x(N(x)&...). Let
T be a theory in L such that for every name c, N(c) is a theorem of T and for
every function symbol f the following is a theorem of T :
∀x 1 ...∀x k ((N(x 1 )& ... & N(x k )) → N( f (x 1 ,...,x k ))).
Show that for any model M of T , the set of a in |M| that satisfies N(x)isthe
domain of an interpretation N such that any sentence S of L is true in N if
and only if its relativization S N is true in M.
17.12 Continuing the preceding series of problem, show that the function assigning
each sentence S of L its relativization S N is a translation. (You may appeal to
Church’s thesis.)
17.13 Consider the interpretation Z of the language {