Undecidable sentences generated by semantic paradoxes

Abstract

In applying the method of arithmetization to a proof of the completeness of the predicate calculus, Bernays has obtained a result which, when applied to set theories formulated in the predicate calculus, may be stated thus.1.1. By adding an arithmetic sentence Con(S) (expressing the consistency of a set theory S) as a new axiom to the elementary number theory Zμ (HB II, p. 293), we can prove in the resulting system arithmetic translations of all theorems of S.It then follows that things definable or expressible in S have images in a simple extension of Zμ, if S is consistent. Since S can be a “strong” system, this fact has interesting consequences. Some of these are discussed by me and some are discussed by Kreisel. Kreisel finds an undecidable sentence of set theory by combining 1.1 and the Cantor diagonal argument. I shall prove below, using similar methods, a few further results, concerned with the notions of truth and designation. The method of numbering sets which I use (see 3.1 below) is different from Kreisel's. While the method used here is formally more elegant, Kreisel's method is much more efficient if we wish actually to calculate the numerical values.