The Relative Consistency of the Axiom of Choice -- Mechanized Using Isabelle/ZF

• Authors: Lawrence C. Paulson
• Venue: CiE '08 Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms
• Year: 2008

Gödel [3] published a monograph in 1940 proving a highly significant theorem, namely that the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are consistent with respect to the other axioms of set theory. This theorem addresses the first of Hilbert's famous list of unsolved problems in mathematics. I have mechanized this work [8] using Isabelle/ZF [5,6]. Obviously, the theorem's significance makes it a tempting challenge; the proof also has numerous interesting features. It is not a single formal assertion, as most theorems are. Gödel [3, p. 33] states it as follows, using Σto denote the axioms for set theory:What we shall prove is that, if a contradiction from the axiom of choice and the generalized continuum hypothesis were derived in Σ, it could be transformed into a contradiction obtained from the axioms of Σalone.Gödel presents no other statement of this theorem. Neither does he introduce a theory of syntax suitable for reasoning about transformations on proofs, surely because he considers it to be unnecessary