Computability and Logic

23.1. ARITHMETICAL DEFINABILITY AND TRUTH 287
the expansion of N that assigns G as denotation the set A. And we say a class of
sets of numbers is arithmetically definable if and only if there is a sentence F(G)of
L G such that is precisely the set of A for which F(G) is true in N G A .
The following two results contrast with Tarski’s theorem to the effect that V is not
arithmetically definable.
23.1 Theorem. For each n, V n is arithmetically definable.
23.2 Theorem. The class {V } whose one and only member is V is arithmetically
definable.
This entire section will be devoted to the proofs. We are going to need certain facts
about recursiveness (or the ‘arithmetization of syntax’):
(1) The set S of code numbers of the sentences of L is recursive.
(2) For each n, the set S n of code numbers of sentences of L with no more than n
occurrences of logical operators is recursive.
(3) There exists a recursive function ν such that if B is a sentence of L with code
number b, then ν(b) is the code number of ∼B.
(4) There exists a recursive function δ such that if B and C are sentences of L with
code numbers b and c, then δ(b, c) is the code number of (B ∨ C).
(5) There exists a recursive function η such that if v is a variable with code numbers q
and F(v) is a formula with code number p, then η(p, q) is the code number of
∃vF(v).
(6) There exists a recursive function σ such that if v, F(v), q, p are as in (5), then for
any m,σ(p, q, m) is the code number of F(m).
(7) The set V 0 of atomic sentences of L that are true in N is recursive.
[We may suppose that ν, δ, η, σ take the value 0 for inappropriate arguments; for
instance, if b is not the code number for a sentence, then ν(b) = 0.]
In every case, intuitively it is more or less clear that the set or function in question is
effectively decidable or computable, so according to Church’s thesis they should all be
recursive. Proofs not depending on appeal to Church’s thesis have been given for (1)
and (3)–(6) in Chapter 15; and the proof for (2) is very similar and could easily have
been included there as well (or placed among the problems at the end of that chapter).
As for (7), perhaps the simplest proof is to note that V 0 consists of the elements
of the recursive set S 0 that are theorems of Q, and equivalently whose negations
are not theorems of Q (since Q proves all true atomic sentences and disproves all
false ones). But we know from Chapter 15 that the set of theorems of Q or any
axiomatizable theory is a semirecursive set, and the set of sentences whose negations
are not theorems is the complement of a semirecursive set. It follows that V 0 is both
a semirecursive set and the complement of one, and is therefore recursive.
The sets above all being recursive, they are definable in arithmetic and indeed in
Q, and the functions are representable. Let S(x), S 0 (x), S 1 (x), S 2 (x),...,Nu(x, y),
Delta(x, y, z), Eta(x, y, z), Sigma(x, y, z, w), and V 0 (x) be defining or representing
formulas for S, S 0 , S 1 , S 2 ,...,ν,δ,η,σ, and V 0 . This machinery will be used in the
proofs of both theorems.