Computability and Logic

SECOND-ORDER LOGIC 283
22.6 Example (Second-order arithmetic). Let P II be the conjunction of the axioms of Q
(as in section 16.2) with the following sentence Ind, called the axiom of induction:
∀X((X0 & ∀x(Xx → Xx ′ )) →∀xXx).
Then an interpretation of the language of arithmetic is a model of P II if and only if it is
isomorphic to the standard interpretation.
Proof: We have already in effect seen in the proof of Example 22.3 that in any
model of Ind, the domain will consist precisely of the denotations of the terms
0, 0 ′ , 0 ′′ ,..., which is to say, of the numerals 0, 1, 2,..., as we usually abbreviate
those terms. We have also seen in section 16.2 that in any model of the axioms
of Q, all the following will be true for natural numbers m, n, and p:
m ≠ n if m ≠ n
m < n if m < n
∼m < n if m ≥ n
m+n=p if m + n = p
m+n≠p if m + n ≠ p
m · n = p if m · n = p
m · n ≠ p if m · n ≠ p.
Now let M be a model of P II . Every element of |M| is the denotation of at least
one m, because M is a model of Ind, and of at most one m, because M is a model
of the axioms of Q and therefore of m ≠ n whenever m ≠ n, by the first fact on the
list above. We can therefore define a function j from |M| to the natural numbers by
letting the value of j for the argument that is the denotation of m by m. By the other
six facts on the list above, j will be an isomorphism between M and the standard
interpretation.
Conversely, P II is easily seen to be true in the standard interpretation, and the proof
of the isomorphism theorem (Proposition 12.5) goes through essentially unchanged
for second-order logic, so any interpretation isomorphic to the standard interpretation
will also be an model of P II .
22.7 Proposition. The compactness theorem fails for second-order logic.
Proof: As in the construction of a nonstandard model of first-order arithmetic, add
a constant c to the language of arithmetic and consider the set
Ɣ ={P II , c ≠ 0, c ≠ 1, c ≠ 2,...}.
Every finite subset Ɣ 0 has a model obtained by expanding the standard interpretation
to assign a suitable denotation to c—any number bigger than all those mentioned in
Ɣ 0 will do. But Ɣ itself does not, because in any model of P II every element is the
denotation of one of the terms 0, 1, 2, and so on.
22.8 Proposition. The (abstract) Gödel completeness theorem fails for second-order
logic: The set of valid sentences of second-order logic is not semirecursive (or even arithmetical).