Computability and Logic

25.3. NONSTANDARD MODELS OF ANALYSIS 313
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 )inM, 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 *ofL*, 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 Löwenheim–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.