Computability and Logic

25.3. NONSTANDARD MODELS OF ANALYSIS 315
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 Gödel 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)ofL** 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 L 1 is contained in a language L 2 , a theory T 1 in L 1 is
contained in a theory T 2 in L 2 , then T 2 is called a conservative extension of T 1 if and
only if every sentence of L 1 that is a theorem of T 2 is a theorem of T 1 . 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 S 0 ∪ S 1 ∪ S 2 ∪···of
a sequence of sets that is defined inductively. In an inductive definition, we specify
a condition F 0 (u) such that u belongs to S 0 if and only if F 0 (u) holds, and specify a
condition F ′ (u, U) such that for all i, u belongs to S i+1 if and only if F ′ (u, S i ) 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 U 0 ,...,U n such that
for all u, u ∈ U 0 , if and only if F 0 (u)
for all i < n, for all u, u ∈ U i+1 if and only if F ′ (u, U i )
x ∈ U n .