CHAPTER 15
Preliminaries It was mentioned in the Introduction that one of the motivations for the development of notions of computability was the following question. Entscheidungsproblem. Given a reasonable set Σ of formulas of a first-order language L and a formula ϕ of L, is there an effective method for determining whether or not Σ ` ϕ? Armed with knowledge of first-order logic on the one hand and of computability on the other, we are in a position to formulate this question precisely and then solve it. To cut to the chase, the answer is usually “no”. G¨odel’s Incompleteness Theorem asserts, roughly, that given any set of axioms in a first-order language which are computable and also powerful enough to prove certain facts about arithmetic, it is possible to formulate statements in the language whose truth is not decided by the axioms. In particular, it turns out that no consistent set of axioms can hope to prove its own consistency. We will tackle the Incompleteness Theorem in three stages. First, we will code the formulas and proofs of a first-order language as numbers and show that the functions and relations involved are recursive. This will, in particular, make it possible for us to define a “computable set of axioms” precisely. Second, we will show that all recursive functions and relations can be defined by first-order formulas in the presence of a fairly minimal set of axioms about elementary number theory. Finally, by putting recursive functions talking about first-order formulas together with first-order formulas defining recursive functions, we will manufacture a self-referential sentence which asserts its own unprovability. Note. It will be assumed in what follows that you are familiar with the basics of the syntax and semantics of first-order languages, as laid out in Chapters 5–8 of this text. Even if you are already familiar with the material, you may wish to look over Chapters 5–8 to familiarize yourself with the notation, definitions, and conventions used here, or at least keep them handy in case you need to check some such point. 111