Computability and Logic

314 NONSTANDARD MODELS
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)ofL*, the sentence
(F(0)&∀x(F(x) → F(x ′ ))) →∀xF(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,...))
→∀xF(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) →∀xx∈ X)
(2) ∀X∀Y (∀x(x ∈ X ↔ x ∈ Y ) → X = Y )
(3) For each formula F(x)ofL*, 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 Gödel theorems
apply to P**), among these are the Gödel 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**.