Computability and Logic

306 NONSTANDARD MODELS
the proof of the model existence lemma in Chapter 13. It is outlined in the problems
at the end of this chapter. While (true) arithmetic is not an axiomatizable theory, P
is, and so the arithmetical Löwenheim–Skolem theorem gives us the following.
25.3 Corollary. There is a nonstandard model of P with domain the natural numbers
in which the denotation of every nonlogical symbol is an arithmetical relation or function.
Proof: As in the proof of the existence of nonstandard models of arithmetic, add a
constant ∞ to the language of arithmetic and apply the compactness theorem to the
theory
P ∪{∞≠ n: n = 0, 1, 2,...}
to conclude that it has a model (necessarily infinite, since all models of P are). The
denotation of ∞ in any such model will be a nonstandard element, guaranteeing that
the model is nonstandard. Then apply the arithmetical Löwenheim–Skolem theorem
to conclude that the model may be taken to have domain the natural numbers, and the
denotations of all nonlogical symbols arithmetical.
The results of the next section contrast sharply with Corollaries 25.2 and 25.3.
25.2 Operations in Nonstandard Models
Our goal in this section is to indicate the proof of two strengthenings of Tennenbaum’s
theorem to the effect that there is no nonstandard model of P with domain the natural
numbers in which the addition and multiplication functions are both recursive, along
with two analogues of these strengthened results. Specifically, the four results are as
follows.
25.4a Theorem. There is no nonstandard model of (true) arithmetic with domain the
natural numbers in which the addition function is arithmetical.
25.4b Theorem (Tennenbaum–Kreisel theorem). There is no nonstandard model of P
with domain the natural numbers in which the addition function is recursive.
25.4c Theorem. There is no nonstandard model of (true) arithmetic with domain the
natural numbers in which the multiplication function is arithmetical.
25.4d Theorem (Tennenbaum–McAloon theorem). There is no nonstandard model of
P with domain the natural numbers in which the multiplication function is recursive.
The proof of Theorem 25.4a will be given in some detail. The modifications
needed to prove Theorem 25.4b and those needed to prove Theorem 25.4c will both
be indicated in outline. A combination of both kind of modifications would be needed
for Theorem 25.4d, which will not be further discussed.
Throughout the remainder of this section, by formula we mean formula and sentence
of the language of arithmetic L, and by model we mean an interpretation of L
with domain the set of natural numbers. For the moment our concern will be with
models of (true) arithmetic. Let M be such a model that is not isomorphic to the