Gödel’s Incompleteness Theorem

Abstract

In this section we show how the syntax of formal languages reduces in principle to arithmetic. We do this by identifying the symbols, expressions, and texts in a finite or countable alphabet A with certain natural numbers (i.e., by numbering them) in such a way that the syntactic operations (juxtaposition, substitution, etc.) are represented by recursive functions, and the syntactic relations (occurrence in an expression, “being a formula,” etc.) are represented by decidable or enumerable sets.

In Chapter II we described how this technique works for Smullyan’s language of arithmetic, but now we shall investigate it more systematically. Our first task is to show that the computability of syntactic operations and the decidability (enumerability) of syntactic relations on the sets of expressions and texts do not depend on how we number them, as long as we adhere to certain weak natural restrictions.

This independence of the method of numbering allows us to consider this numbering not only as a technical device, but also as a reflection of a deep equivalence between arithmetic and the combinatorial properties of formal texts. In modern computers, where a single store-location may serve consecutively as a number, a name (code), and a command, this equivalence between syntax and arithmetic is realized “in the flesh” and is accepted as a basic principle. This was not the case, however, in 1931, when Gödel first introduced the concept of numbering.