Computability and Logic

23
Arithmetical Definability
Tarski’s theorem tells us that the set V of (code numbers of) first-order sentences of the
language in arithmetic that are true in the standard interpretation is not arithmetically
definable. In section 23.1 we show that this negative result is poised, so to speak, between
two positive results. One is that for each n the set V n of sentences of the language of
arithmetic of degree of complexity n that are true in the standard interpretation is
arithmetically definable (in a sense of degree of complexity to be made precise). The
other is that the class {V} of sets of natural numbers whose one and only member is V
is arithmetically definable (in a sense of arithmetical definability for classes to be made
precise). In section 23.2 we take up the question whether the class of arithmetically
definable sets of numbers is an arithmetically definable class of sets. The answer is
negative, according to Addison’s theorem. This result is perhaps most interesting on
account of its method of proof, which is a comparatively simple application of the method
of forcing originally devised to prove the independence of the continuum hypothesis in
set theory (as alluded to in the historical notes to Chapter 18).
23.1 Arithmetical Definability and Truth
Throughout this chapter we use L and N for the language of arithmetic and its
standard interpretation (previously called L* and N *), and V for the set of code
numbers of first-order sentences of L ture in N . It will be convenient to work with
a version of logic in which the only operators are ∼ and ∨ and ∃ (& and ∀ being
treated as unofficial abbreviations). We measure the ‘complexity’ of a sentence by
the number of occurrences of logical operators ∼ and ∨ and ∃ in it. (Our results
do, however, go through for other reasonable notions of measures of complexity: see
the problems at the end of the chapter). By V n we mean the set of code numbers of
first-order sentences of L of complexity ≤n that are true in N .
We are going to be discussing natural numbers, sets of natural numbers, and sets
of sets of natural numbers. To keep the levels straight, we generally use numbers for
the natural numbers, sets for the sets of natural numbers, and classes for the sets of
sets of natural numbers. We write L c for the expansion of L by adding a constant c,
and Na c for the expansion of N that assigns c as denotation the number a. Then a
set S of numbers is arithmetically definable if and only if there is a sentence F(c)of
L c such that S is precisely the set of a for which F(c) is true in Na c. Analogously,
we write L G for the expansion of L by adding a one-place predicate G, and NA G for
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