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.