Computability and Logic

25
Nonstandard Models
By a model of (true) arithmetic is meant any model of the set of all sentences of the
language L of arithmetic that are true in the standard interpretation N .Byanonstandard
model is meant one that is not isomorphic to N . The proof of the existence of an
(enumerable) nonstandard model of arithmetic is as an easy application of the compactness
theorem (and the Löwenheim–Skolem theorem). Every enumerable nonstandard
model is isomorphic to a nonstandard model M whose domain is the same as that of
N , namely, the set of natural numbers; though of course such an M cannot assign
the same denotations as N to the nonlogical symbols of L. In section 25.1 we analyze
the structure of the order relation in such a nonstandard model. A consequence of this
analysis is that, though the order relation cannot be the standard one, it at least can be
a recursive relation. By contrast, Tennenbaum’s theorem tells us that it cannot happen
that the addition and multiplication relations are recursive. This theorem and related
results will be taken up in section 25.2. Section 25.3 is a sort of appendix (independent
of the other sections, but alluding to results from several earlier chapters) concerning
nonstandard models of an expansion of arithmetic called analysis.
25.1 Order in Nonstandard Models
Let M be a model of (true) arithmetic not isomorphic to the standard model N . (The
existence of such models was established in the problems at the end of Chapter 12, as
an application of the compactness theorem.) What does such a model look like? We’ll
call the objects in the domain |M| NUMBERS. M assigns as denotation to the symbol
0 some NUMBER O we’ll call ZERO, and to the symbol ′ some function † on NUMBERS
we’ll call SUCCESSOR. It assigns to < some relation ≺ we’ll call LESS THAN, and
to + and · some functions ⊕ and ⊗ we’ll call ADDITION and MULTIPLICATION. Our
main concern in this section will be to understand the LESS THAN relation.
First of all, no NUMBER is LESS THAN itself. For no (natural) number is less than
itself. So ∀x ∼ x < x is true in N , so it is true in M, and so as asserted no NUMBER
is LESS THAN itself. This argument illustrates our main technique for obtaining information
about the ‘appearance’ of M: observe that the natural numbers have a certain
property, conclude that a certain sentence of L is true in N , infer that it must also be
true in M (since the same sentences of L are true in M as in N ), and decipher the
sentence ‘over’ M. In this way we can conclude that exactly one of any two NUMBERS
is LESS THAN the other, and that if one NUMBER is LESS THAN another, which is LESS
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