312 NONSTANDARD MODELS

And suppose conversely 2y is a wth power, say 2y = tw . Then t cannot be divisible by any odd prime, and so must be a power of 2, say t = 2u . Then 2y = (2u )w = 2uw , and y = uw, so y 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 x F(x) is true in an interpretation if every element of the lower domain satisfies F(x), while a sentence X G(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 Lowenheim¨ – Skolem theorem (including the strong Lowenheim¨ –Skolem theorem of Chapter 19). Note that in the Lowenheim¨ –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* of L. 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

nonlogical symbol of L* the same denotation as does the given interpretation. Thus the lower part of N** is N*. A sentence of L* will be true in an interpretation of L** if and only if it is true in the lower part of that interpretation. Thus a sentence of L* is a theorem of (that is, is in) true arithmetic if and only if it is a theorem of true analysis.

Our first aim in this section will be to establish the existence of nonstandard models of analysis of two distinct kinds. An interpretation of L** is called an -model if (as in the standard interpretation) the elements of the upper domain are sets of elements of the lower domain, and the interpretation of is the membership or elementhood relation (between elements of the lower and the upper domain). The sentence

X Y ( x(x X ↔ x Y ) → X = Y )

is called the axiom of extensionality. Clearly it is true in any -model and hence in any model isomorphic to an -model. Conversely, any model M of extensionality is isomorphic to an -model M#. [To obtain M# from M, keep the same lower domain and the same interpretations for symbols of L*, replace each element α of the upper domain of M by the set α# of all elements a of the lower domain such that a M α, and interpret not as the relation M but as . The identity function on the lower domain together with the function sending α to α# is an isomorphism. The only point that may not be immediately obvious is that the latter function is one-to-one. To see this, note that if α# = β#, then α and β satisfy x(x X ↔ x Y ) in M, and since (2) is true in M, α and β must satisfy X = Y , that is, we must have α = β.] Since we are going to be interested only in models of extensionality, we may restrict our attention to -models.

If the lower part of an -model M is the standard model of arithmetic, we call M an ω-model. The standard model of analysis is, of course, an ω-model. If an ω-model of analysis is nonstandard, its upper domain must consist of some class of sets properly contained in the class of all sets of numbers. If the lower part of an-model M is isomorphic to the standard interpretation N* of L*, then M as a whole is isomorphic to an ω-model M#. [If j is the isomorphism from N* to the lower part of M, replace each element α of the upper domain of M by the set of n such that j(n) α, to obtain M#.] So we may restrict our attention to models that are of one of two kinds, namely, those that either are ω-models, or have a nonstandard lower part.

Our first result is that nonstandard models of analysis of both kinds exist.

25.9 Proposition. Both nonstandard models of analysis whose lower part is a nonstandard model of arithmetic and nonstandard ω-models of analysis exist.

Proof: The existence of nonstandard models of arithmetic was established in the problems at the end of Chapter 12 by applying the compactness theorem to the theory that results upon adding to arithmetic a constant ∞ and the sentences ∞ = n for all natural numbers n. The same proof, with analysis in place of arithmetic, establishes the existence of a nonstandard model of analysis whose lower parts is a nonstandard model of arithmetic. The strong Lowenheim¨ –Skolem theorem implies the existence of an enumerable subinterpretation of the standard model of analysis that is itself a model of analysis. This must be an ω-model, but it cannot be isomorphic to the standard model, whose upper domain is nonenumerable.

The axiomatizable theory in L* to which logicians have devoted the most attention is P, which consists of the sentences deducible from the following axioms:

(0)The finitely many axioms of Q

(1)For each formula F(x) of L*, the sentence

(F(0) & x(F(x) → F(x ))) → x F(x).

It is to be understood that in (1) there may be other free variables u, v, . . . present, and that what is really meant by the displayed expression is the universal closure

u v · · · (F(0, u, v, . . .) & x(F(x, u, v, . . .) → F(x , u, v, . . .)) → x F(x, u, v, . . .)).

The sentence in (1) is called the induction axiom for F(x).

The axiomatizable theory in L** to which logicians have devoted the most attention is the theory P** consisting of the sentences deducible from the following axioms:

(0) The finitely many axioms of Q

(1*) X(0 X & x(x X → x X) → x x X)

(2)X Y ( x(x X ↔ x Y ) → X = Y )

(3)For each formula F(x) of L*, the sentence

X x(x X ↔ F(x)).

It is to be understood that in (3) there may be other free variables u, v, . . . and/or U, V, . . . present, and that what is really meant by the displayed expression is the universal closure

u v · · · U V · · · X x(x X ↔ F(x, u, v, . . . , U, V, . . .)).

The sentence (1*) is called the induction axiom of P**, the extensionality axiom

(2) has already been encountered, and the sentence (3) is called the comprehension axiom for F(x). We call P** axiomatic analysis.

Since the set of theorems of (true) arithmetic is not arithmetical, the set of theorems of (true) analysis is not arithmetical, and a fortiori is not semirecursive. By contrast, the set of theorems of axiomatic analysis P** is, like the set of theorems of any axiomatizable theory, semirecursive. There must be many theorems of (true) analysis that are not theorems of axiomatic analysis, and indeed (since the Godel¨ theorems apply to P**), among these are the Godel¨ and Rosser sentences of P**, and the consistency sentence for P**.

Note that the induction axiom (1) of P for F(x) follows immediately from the induction axiom (1) of P** together with the comprehension axiom (3) for F(x). Thus every theorem of P is a theorem of P**, and the lower part of any model of P** is a model of P. We say a model of P is expandable to a model of P** if it is the lower part of a model of P**. Our second result is to establish the nonexistence of certain kinds of nonstandard models of P**.

25.10 Proposition

Not every model of P can be expanded to a model of P**.

Proof: We are not going to give a full proof, but let us indicate the main idea. Any model of P that can be expanded to a model of P** must be a model of every sentence of L* that is a theorem of P**. Let A be the consistency sentence for P (or the Godel¨ or Rosser sentence). Then A is not a theorem of P, and so there is a model of P { A}. We claim such a model cannot be expanded to a model of P**, because A is provable in P**. The most simple-minded proof of the consistency of P is just this: every axiom of P is true, only truths are deducible from truths, and 0 = 1 is not true; hence 0 = 1 is not deducible from P. In section 23.1 we in effect produced a formula F(X) of L** which is satisfied in the standard model of analysis by and only by the set code numbers of sentences of L* that are true in the lower part of that model (that is, in the standard model of arithmetic). Working in P**, we can introduce the abbreviation True(x) for X(F(X) & x X), and ‘formalize’ the simple-minded argument just indicated. (The work of ‘formalization’ required, which we are omitting, is extensive, though not so extensive as would be required for a complete proof of the second incompleteness theorem.)

Recall that if a language L1 is contained in a language L2, a theory T1 in L1 is contained in a theory T2 in L2, then T2 is called a conservative extension of T1 if and only if every sentence of L1 that is a theorem of T2 is a theorem of T1. What is shown in the proof indicated for the preceding proposition is, in this terminology, that P** is not a conservative extension of P.

A weaker variant P+ allows the comprehension axioms (3) only for formulas F(x) not involving bound upper variables. [There may still be, in addition to free lower variables u, v, . . . , free upper variables U, V, . . . in F(X ).] P+ is called (strictly) predicative analysis. When one specifies a set by specifying a condition that is necessary and sufficient for an object to belong to the set, the specification is called impredicative if the condition involves quantification over sets. Predicative analysis does not allow impredicative specifications of sets. In ordinary, unformalized mathematical argument, impredicative specifications of sets of numbers are comparatively common: for instance, in the first section of the next chapter, an ordinary, unformalized mathematical proof of a principle about sets of natural numbers called the ‘infinitary Ramsey’s theorem’ will be presented that is a typical example of a proof that can be ‘formalized’ in P** but not in P+.

An innocent-looking instance of impredicative specification of a set is implicitly involved whenever we define a set S of numbers as the union S0 S1 S2 · · · of a sequence of sets that is defined inductively. In an inductive definition, we specify a condition F0(u) such that u belongs to S0 if and only if F0(u) holds, and specify a condition F (u, U ) such that for all i, u belongs to Si+1 if and only if F (u, Si ) holds. Such an inductive definition can be turned into a direct definition, since x S if and only if

there exists a finite sequence of sets U0, . . . , Un such that for all u, u U0, if and only if F0(u)

for all i < n, for all u, u Ui+1 if and only if F (u, Ui ) x Un .

But while the quantification ‘there exists a finite sequence of sets’ can by suitable coding be replaced by a quantification ‘there exists a set’, in general the latter quantification cannot be eliminated. The inductive definition implicitly involves—what the corresponding direct definition explicitly involves—an impredicative specification of a set. In general, one cannot ‘formalize’ in P+ arguments involving this kind of inductive specification of sets, even if the conditions F0 and F involve no bound upper variables.

Also, one cannot ‘formalize’ in P+ the proof of the consistency sentence for P indicated in the proof of the preceding proposition. [One can indeed introduce the abbreviation True(x) for X(F(X) & x X ), but one cannot in P prove the existence of {x: True(x)}, and so cannot apply the induction axiom to prove assertions involving the abbreviation True(x).] So the proof indicated for the preceding proposition fails for P+ in place of P*. In fact, not only is the consistency sentence for P not an example of a sentence of L* that is a theorem of P+ and not of P, but actually there can be no example of such sentence: P+ is a conservative extension of P.

Our last result is a proposition immediately implying the fact just stated.

25.11 Proposition. Every model of P can be expanded to a model of P+.

Proof: Let M be a model of P. Call a subset S of the domain |M| parametrically definable over M if there exist a formula F(x, y1, . . . , ym ) of L* and elements a1, . . . , am of |M| such that

S = {b: M |= F[b, a1, . . . , am ]}.

Expand M to an interpretation of L** by taking as upper domain the class of all parametrically definable subsets of M, and interpreting as .We claim the expanded model M+ is a model of P+. The axioms that need checking are induction (1) and comprehension (3) (with F having no bound upper variables). Leaving the former to the reader, we consider an instance of the latter:

u1 u2 U1 U2 X x(x X ↔ F(x, u1, u2, U1, U2)).

(In general, there could be more than two us and more than two U s, but the proof would be no different.) To show the displayed axiom is true in M+, we need to show that for any elements s1, s2 of |M| and any parametrically definable subsets S1, S2 of |M| there is a parametrically definable subset T of |M| such that

M+ |= x(x X ↔ F(x, u1, u2, U1, U2))[s1, s2, S1, S2, T ].

Equivalently, what we must show is that for any such s1, s2, S1, S2, the set

T = {b: M+ |= F(x, u1, u2, U1, U2)[s1, s2, S1, S2, b]}

is parametrically definable. To this end, consider parametric definitions of U1, U2:

U1 = {b: M |= G1[b, a11, a12]}

U2 = {b: M |= G2[b, a21, a22]}.