Journal of Automated Reasoning

, Volume 17, Issue 3, pp 291–323 | Cite as

Mechanizing set theory

Cardinal arithmetic and the Axiom of Choice
  • Lawrence C. Paulson
  • Krzysztof Grabczewski


Fairly deep results of Zermelo-Frænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplications is κ ⊗ κ=κ, where κ 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, and involves highly technical material. The definitions used in the proofs are largely faithful in style to the original mathematics.

Key words

Isabelle cardinal arithmetic Axiom of Choice set theory QED project 


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Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Lawrence C. Paulson
    • 1
  • Krzysztof Grabczewski
    • 2
  1. 1.Computer LaboratoryUniversity of CambridgeUK
  2. 2.Nicholas Copernicus UniversityToruńPoland

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