Computability and Logic

25.2. OPERATIONS IN NONSTANDARD MODELS 307
standard model N , and let us use ⊕ and ⊗ as in the preceding section for the denotations
it assigns to the addition and multiplication symbols.
A notational preliminary: Our usual notation for the satisfaction in a model M of
a formula F(x, y) by elements a and b of the domain has been M |= F[a, b]. For the
remainder of this chapter, rather than write, for instance, ‘let F(x, y) be the formula
∃zx = y · z and let M |= F[a, b]’, we are just going to write ‘let M |= ∃zx =
y · z[a, b]’. (Potentially this briefer notation is ambiguous where there is more than
one free variable, since nothing in it explicitly indicates that it is a that goes with x and
b with y rather than the other way around; actually, context and alphabetical order
should always be sufficient to indicate what is intended.) Thus instead of writing
‘Let F(z) be the formula n < z and suppose M |= F[d]’, we just write ‘Suppose
M |= n < z[d]’. In this notation, a number d is a nonstandard element of M if and
only if for for every n, M |= n < z[d]. (If d is nonstandard, M |= d < z[d].)
We know from the previous section that nonstandard elements exist. The key to
proving Theorem 25.4a is a rather surprising result (Lemma 25.7a below) asserting
the existence of nonstandard elements with special properties. In the statement of this
result and the lemmas needed to prove it, we write π(n) for the nth prime (counting 2
as the zeroth, 3 as the first, and so on). In order to be able to write about π in
the language of arithmetic, fix a formula (x, y) representing the function π in Q
(and hence in P and in arithemetic), as in section 16.2. Also, abbreviate as x | y
the rudimentary formula defining the relation ‘x divides y’. Here, then, are the key
lemmas.
25.5a Lemma. Let M be a nonstandard model of arithmetic. For any m > 0,
M |= ∀x m · x = x + ···+ x
(m xs).
25.6a Lemma. Let M be a nonstandard model of arithmetic. Let A(x) be any formula
of L. Then there is a nonstandard element d such that
M |= ∃ y∀x < z (∃w((x,w)&w | y) ↔ A(x))[d].
25.7a Lemma. Let M be a nonstandard model of arithmetic. Let A(x) be any formula
of L. Then there exists a b such that for every n,
M |= A(n) if and only if for some a, b = a ⊕···⊕a [π(n) as].
Proof of Lemma 25.5a: The displayed sentence is true in N , and hence is true in
M.
Proof of Lemma 25.6a: It is enough to show that the sentence
∀z∃y∀x < z (∃w((x,w)&w | y) ↔ A(x))
is true in N , since it must then be true in M. Now what this sentence says, interpreted
over N , is just that for every z there exists a positive y such that for all x < z, the xth
prime divides y if and only if A(x) holds. It is enough to take for y the product of the
xth prime for all x < z such that A(x) holds.