Computability and Logic

284 SECOND-ORDER LOGIC
Proof: A first-order sentence A of the language of arithmetic is true in the standard
interpretation if and only if it is true in all interpretations isomorphic to the standard
one, and hence by the preceding example if and only if it is true in all models of
P II , or equivalently, if and only if P II → A is valid. The function taking (the code
number of) a first-order sentence A to (the code number of) the second-order sentence
P II → A is clearly recursive. (Compare the proof of Theorem 17.6.) Hence if the set of
(code numbers of) valid second-order sentences were semirecursive, the set of (code
numbers of) sentences of the language of arithmetic true in the standard interpretation
would be also. But the latter set is not arithmetical (by Theorem 17.3) and a fortiori
not semirecursive.
Proposition 22.8 is sometimes formulated as follows: ‘Second-order logic is
incomplete’. A more accurate formulation would be: ‘No sound proof procedure
for second-order logic is complete’. (After all, it’s not the logic that’s incomplete, but
candidate proof procedures.)
We conclude this chapter with a preview of the next. Recall that a set S of natural
numbers is arithmetically definable, or simply arithmetical, if there is a first-order
formula F(x) of the language of arithmetic such that S consists of just those m for
which F(m) is true in the standard interpretation, or equivalently, just those m that
satisfy F(x) in the standard interpretation. A set S of natural numbers is analytically
definable or analytical if there is a first- or second-order formula φ(x) of the language
of arithmetic such that S consists of just those m that satisfy φ(x) in the standard
interpretation. Let us, for the space of this discussion, use the word class for sets of sets
of natural numbers. Then a class of sets of natural numbers is arithmetical if there
is a second-order formula F(X) with no bound relation or function variables such that
consists of just those sets M that satisfy F(X) in the standard intepretation. A class
of sets of natural numbers is analytical if there is a second-order formula φ(X) such
that consists of just those sets M that satisfy φ(X) in the standard interpretation.
We have seen that recursive and semirecursive sets are arithmetical, but that the set of
(code numbers of) first-order sentences of the language of arithmetic that are true in
the standard interpretation is not arithmetical. It can similarly be shown that the set of
first- and second-order sentences true in the standard interpretation is not analytical.
However, the set V of (code numbers of) first-order sentences true in the standard
interpretation is analytical. This follows from the fact, to be proved in the next chapter,
that the class {V } of sets of natural numbers whose one and only member is the set
V is arithmetical. The latter result means that there is a second-order formula F(X)
with no bound relation or function variables such that V is the one and only set that
satisfies F(X) in the standard interpretation. From this it follows that V is precisely
the set of m that satisfy ∃X(F(X)&Xx); and this shows that, as asserted, V is
analytical. It will also be shown that the class of arithmetical sets of natural numbers
is not arithmetical. (Again, this class can be shown to be analytical.) In order to keep
the next chapter self-contained and independent of this one, a different definition of
arithmetical class will be given there, not presupposing familiarity with second-order
logic. However, the reader who is familiar with second-order logic should have no
difficulty recognizing that this definition is equivalent to the one given here.