Mechanising Set Theory: Cardinal Arithmetic and the Axiom of Choice

Lawrence C. Paulson, University of Cambridge, England
Krzysztof Grąbczewski, Nicholas Copernicus University, Poland

Fairly deep results of Zermelo-Fraenkel (ZF) set theory have been mechanised using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is K*K=K, where K is any infinite cardinal. Proving this result required developing theories of orders, order-isomorphisms, order types, ordinal arithmetic, cardinals, etc.; this covers most of Kunen, Set Theory, Chapter I.

Furthermore, we have proved the equivalence of 7 formulations of the Well-ordering Theorem and 20 formulations of AC; this covers the first two chapters of Rubin and Rubin, Equivalents of the Axiom of Choice. The definitions used in the proofs are largely faithful in style to the original mathematics.