This chapter falls into three parts. We first introduce Gödel's simple but wonderfully powerful idea of associating the expressions of a formal theory with code numbers. In particular, we'll fix on a scheme for assigning code numbers first to expressions of LA and then to proof-like sequences of expressions. This double coding scheme will correlate various syntactic properties with purely numerical properties.

For example, take the syntactic property of being a term of LA. We can define a corresponding numerical property Term, where Term(n) holds just when n codes for a term. Likewise, we can define Atom(n), Wff(n), and Sent(n) which hold just when n codes for an atomic wff, a wff, or a closed wff (sentence) respectively. It will be easy to see that these numerical properties are primitive recursive ones.

More excitingly, we can define the numerical relation Prf (m, n) which holds just when m is the code number in our scheme of a PA-derivation of the sentence with number n. It will also be easy to see – at least in an informal way – that this relation too is primitive recursive.

The second part of the chapter introduces the idea of the diagonalization of a wff. This is the idea of taking a wff ϕ(y), and substituting (the numeral for) its own code number in place of the free variable.