Mappings and the arithmetization of meta-mathematics

In an earlier post, we discussed mappings in general and then described Gödel numbering, which is a mapping between the set of well-formed formulas (wff's) in a formal system and the set of positive integers.

In part B of section VII, Nagel and Newman describe a more interesting mapping used in Gödel's proof. In this mapping, the domain is the set of meta-mathematical statements about structural properties of wff's, while the co-domain is the set of wff's. This post describes how this mapping enabled Gödel to arithmetize meta-mathematics.

Mappings and Gödel numbering

Section VI of Nagel and Newman's book describes mappings. A mapping is an operation that applies to two sets called the domain and the co-domain. More specifically, a mapping associates to each element of the domain one or more elements of the co-domain.