> Gödel began by devising a method that would transform statements about the process of proof into statements about the properties of numbers. Doing so would al

> Gödel began by devising a method that would transform statements about the process of proof into statements about the properties of numbers. Doing so would allow him to write metamathematical assertions about the construction of proofs as simple arithmetical formulas, which could thus be expressed within the syntax of a mathematical system itself. Turning metamathematics into simple mathematical propositions was the first of his self-referential feats of sleight of hand. The scheme he devised made it possible to convert any string of symbols used to express a mathematical or logical formula in Principia Mathematica into a single, unique integer. Russell and Whitehead had used a deliberately limited set of a dozen “primitive” symbols in setting up their system, and they showed that with these symbols any proposition of logic or number theory could be expressed, albeit in an often extraordinarily cumbersome fashion. Gödel first assigned a numerical value to each symbol. He used several different coding systems in his different presentations of the Incompleteness Theorem; in the series of lectures he gave in Princeton in 1934 he used the scheme:25 (For definitions of these symbols and a fuller explanation of Gödel’s proof, see appendix.) The particularly clever part of the coding scheme is that it is reversible. By using powers of prime factors to generate the integer representing a string of symbols in a complete formula such as “x = 0,” the resulting formula number, its “f-number” Gödel called it, can be “decoded” through a straightforward arithmetic process back to the statement x = 0. Really nice book. It contains a very good description of arithmetization.