Computability and Logic

312 NONSTANDARD MODELS
And suppose conversely 2 y is a wth power, say 2 y = t w . Then t cannot be divisible
by any odd prime, and so must be a power of 2, say t = 2 u . Then 2 y = (2 u ) w = 2 uw ,
and y = uw, soy is divisible by w.
25.3 Nonstandard Models of Analysis
In the language L* of arithmetic, under its standard interpretation N * (to revert to
our former notation), we can directly ‘talk about’ natural numbers, and can indirectly,
through coding, ‘talk about’ finite sets of natural numbers, integers, rational numbers,
and more. We cannot, however, ‘talk about’ arbitrary sets of natural numbers or
objects that might be coded by these, such as real or complex numbers. The language
of analysis L**, and its standard interpretation N **, let us do so.
This language is an example of a two-sorted first-order language. In two-sorted
first-order logic there are two sorts of variables: a first sort x, y, z,..., which may
be called lower variables, and a second sort X, Y, Z,...,which may be called upper
variables. For each nonlogical symbol of a two-sorted language, it must be specified
not only how many places that symbol has, but also which sorts of variables go into
which places. An interpretation of a two-sorted language has two domains, upper
and lower. A sentence ∀xF(x) is true in an interpretation if every element of the
lower domain satisfies F(x), while a sentence ∀XG(X) is true if every element of
the upper domain satisfies G(X). Otherwise the definitions of language, sentence,
formula, interpretation, truth, satisfaction, and so forth are unchanged from ordinary
or one-sorted first-order logic.
An isomorphism between two interpretations of a two-sorted language consists of
a pair of correspondences, one between the lower domains and the other between
the upper domains of the two interpretations. The proof of the isomorphism lemma
(Proposition 12.5) goes through for two-sorted first-order logic, and so do the proofs
of more substantial results such as the compactness theorem and the Löwenheim–
Skolem theorem (including the strong Löwenheim–Skolem theorem of Chapter 19).
Note that in the Löwenheim–Skolem theorem, an interpretation of a two-sorted language
counts as enumerable only if both its domains are enumerable.
In the language of analysis L** the nonlogical symbols are those of L*, which
take only lower variables, plus a further two-place predicate ∈, which takes a lower
variable in its first place but an upper in its second. Thus x ∈ Y is an atomic formula,
but x ∈ y, X ∈ Y , and X ∈ y are not. In the standard interpretation N ** of L*, the
lower domain is the set of natural numbers and the interpretation of each symbol of
L is the same as in the standard interpretation N *ofL. The upper domain is the
class of all sets of natural numbers, and the interpretation of ∈ is the membership or
elementhood relation ∈ between numbers and sets of numbers. As (true) arithmetic
is the set of sentences of L* true in N *, so (true) analysis is the set of all sentences
of L** true in N **. A model of analysis is nonstandard if it is not isomorphic to
N **. Our aim in this section is to gain some understanding of nonstandard models
of (true) analysis and some important subtheories thereof.
By the lower part of an interpretation of L**, we mean the interpretation of L*
whose domain is the lower domain of the given interpretation, and that assigns to each