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