Computability and Logic

288 ARITHMETICAL DEFINABILITY
Proof of Theorem 23.1: A sentence A that contains n + 1 logical operators is
either the negation ∼B of a sentence B containing n operators, or the disjunction
B ∨ C of two sentences B and C each containing at most n operators, or else an
existential quantification ∃vF(v) of a formula F(v) containing n operators. In the last
case, for each m, the sentence F(m) also contains n operators. In the first case A will
be true if and only if B is not true. In the second case, A will be true if and only if B
is true or C is true. In the third case, A will be true if and only if, for some m, F(m)
is true.
In terms of V n , we can therefore characterize V n+1 as the set of those numbers a
in S n+1 such that either k is in V n ; or for some b, a = ν(b) and b is not in V n ;orfor
some b and c, a = δ(b, c) and either b is in V n or c is in V n ; or finally, for some p
and q, a = η(p, q), and for some m,σ(p, q, m) isinV n .SoifV n (x) arithmetically
defines V n , the following formula V n+1 (x) arithmetically defines V n+1 :
S n+1 (x)&{V n (x) ∨∃y[Nu(y, x)&∼V n (y)]
∨∃y∃z[Delta(y, z, x)&(V n (y) ∨ V n (z))]
∨∃y∃z[Eta(y, z, x)&∃u∃w(Sigma(y, z, u, w)&V n (w))]}.
Since we know V 0 is arithmetically definable, it follows by induction that V n is
arithmetically definable for all n.
Proof of Theorem 23.2: The set of sentences true in N can be characterized as
the unique set Ɣ such that:
Ɣ contains only sentences of L.
For any atomic sentence A, A is in Ɣ if and only if A is a true atomic sentence.
For any sentence B, ∼B is in Ɣ if and only if B is not in Ɣ.
For any sentences B and C, (B ∨ C)isinƔ if and only if B is in Ɣ or C is in Ɣ.
For any variable v and formula F(v), ∃vF(v)isinƔ if and only if, for some m,
F(m)is in Ɣ.
The set V of code numbers of sentences true in N can therefore be characterized
as the unique set M such that:
For all b, if b is in M, then b is in S.
For all a, if a is in S 0 , then a is in M if and only if a is in V 0 .
For all b, if ν(b) isinS, then ν(b)isinM if and only if b is not in M.
For all b and c, if δ(b, c)isinS, then δ(b, c)isinM if and only if either b is in
M or c is in M.
For all p and q, ifη(p, q) isinS, then η(p, q) isinM if and only if, for some
m,σ(p, q, m) isinM.
So on expanding L be adding the one-place predicate G, ifweletF(G) be the
conjunction
∀x(Gx → S(x)) &
∀x(S 0 (x) → (Gx ↔ V 0 (x))) &
∀x∀y((Nu(x, y)&S(y)) → (Gy ↔∼Gx)) &
∀x∀y∀z((Delta(x, y, x)&S(z)) → (Gz ↔ (Gx ∨ Gy))) &
∀x∀y∀z((Eta(x, y, z)&S(z)) → (Gz ↔∃u∃w(Sigma(x, y, u, w)&Gw)))