In Chapter 3, we proved that the theorems of any properly axiomatized theory – and hence, in particular, the theorems of any properly axiomatized arithmetic – can be effectively enumerated. In this chapter, we prove by contrast that the truths of any sufficiently expressive arithmetic language can't be effectively enumerated (we will explain in just a moment what ‘sufficiently expressive’ means).

Suppose then that T is a properly axiomatized theory with a sufficiently expressive language. Since T is axiomatized, its theorems can be effectively enumerated. Since T's language is sufficiently expressive, the truths of its language can't be effectively enumerated. Hence the theorems and the truths can't be the same: either some T-theorems aren't truths, or some truths aren't T-theorems. Let's concentrate on sound theories whose theorems are all true. Then for any sound axiomatized theory T which is sufficiently expressive, there will be truths which aren't T-theorems. Let ϕ be such an unprovable truth: then ¬ϕ will be false, so that too will be unprovable in our sound theory T. Hence T must be negation incomplete.

So much for the headline news. The rest of this chapter fills in the details.

Sufficiently expressive languages

Recall: a two-place relation R is effectively decidable iff there is an algorithmic procedure that decides whether Rmn, for any given m and n (Section 2.2). And a relation R can be expressed in language L iff there is an open L-wff ϕ such that is true iff Rmn (Section 4.5).