Computability and Logic

25.1. ORDER IN NONSTANDARD MODELS 305
arithmetic is isomorphic to the ordering < K of K. Hence the order relations in any
two such models are isomorphic to each other.
This result can be extended from models of (true) arithmetic to models of the
theory P (introduced in Chapter 16).
25.1b Theorem. The order relations on any two enumerable nonstandard models of P
are isomorphic.
Proof: We indicate the proof in outline. What one needs to do in order to extend
Theorem 25.1a from models of arithmetic to models of P is to replace every argument
‘S must be true in M because S is true in N ’ that occurs above, by the argument ‘S
must be true in M because S is a theorem of P’. To show that S is indeed a theorem
of P, one needs to ‘formalize’ in P the ordinary, unformalized mathematical proof
that S is true in N . In some cases (for instance, laws of arithmetic) this has been done
already in Chapter 16; in the other cases (for instance, the existence of averages) what
needs to be done is quite similar to what was done in Chapter 16. Details are left to
the reader.
Any enumerable model of arithmetic or P (or indeed any theory) is isomorphic
to one whose domain is the set of natural numbers. Our interest in the remainder of
this chapter will be in the nature of the relations and functions that such a model
assigns as denotations to the nonlogical symbols of the language. A first result on
this question is a direct consequence of Theorem 25.1a.
25.2 Corollary. There is a nonstandard model of arithmetic with domain the natural
numbers in which the order relation is a recursive relation (and the successor function a
recursive function).
Proof: We know the order relation on any nonstandard model of arithmetic is
isomorphic to the order < K on the set K defined in the proof of Theorem 25.1a. The
main step in the proof of the corollary will be relegated to the problems at the end of the
chapter. It is to show that there is a recursive relation ≺ on the natural numbers that is
also isomorphic to the order < K on the set K . Now, given any enumerable nonstandard
model M of arithmetic, there is a function h from the natural numbers to |M| that
is an isomorphism between the ordering ≺ on the natural numbers and the ordering
< M on M. Much as in the proof of the canonical-domains lemma (Corollary 12.6),
define an operation † on natural numbers by letting n † be the (unique) m such that
h(m) = h(n) ′M ; and define functions ⊕ and ⊗ similarly. Then the interpretation with
domain the natural numbers and with ≺, † , ⊕, ⊗ as the denotation of