LMS Journal of Computation and Mathematics

Research Article

The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle⁄zf

Lawrence C. Paulsona1

a1 Computer Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 OFD, England lcp@cl.cam.ac.uk, http://www.cl.cam.ac.uk/users/lcp

Abstract

The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle⁄ZF, building on a previous mechanization of the reflection theorem. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kenneth Kunen's Set theory: an introduction to independence proofs, and could support the formalization of further material from that book. It also serves as an example of what to expect when deep mathematics is formalized.

(Received January 07 2003)

(Revised September 04 2003)

(Accepted October 13 2003)