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Set theory in first-order logic: Clauses for Gödel's axioms

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Abstract

In this paper we present a set of clauses for set theory, thus developing a foundation for the expression of most theorems of mathematics in a form acceptable to a resolution-based automated theoren prover. Because Gödel's formulation of set theory permits presentation in a finite number of first-orde formulas, we employ it rather than that of Zermelo-Fraenkel. We illustrate the expressive power of thi formulation by providing statements of some well-known open questions in number theory, and give some intuition about how the axioms are used by including some sample proofs. A small set of challeng problems is also given.

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Authors and Affiliations

  1. Microelectronics and Computer Technology Corporation, 9430 Research Boulevard, 78759, Austin, TX, USA

    Robert Boyer

  2. Mathematics and Computer Science Division, Argonne National Laboratory, 60439 4844, Argonne, IL, USA

    Ewing Lusk, William McCune, Ross Overbeek & Lawrence Wos

  3. Artificial Intelligence Center, SRI International, 94025, Menlo Park, CA, USA

    Mark Stickel

Authors
  1. Robert Boyer
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  2. Ewing Lusk
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  3. William McCune
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  4. Ross Overbeek
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  5. Mark Stickel
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  6. Lawrence Wos
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Additional information

This work was partially supported by the Applied Mathematical Sciences subprogram of the office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38, and partially supported by the Defence Advanced Research Projects Agency under contract N00039-84-K-0078 with the Naval Electronic Systems Command.

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Boyer, R., Lusk, E., McCune, W. et al. Set theory in first-order logic: Clauses for Gödel's axioms. J Autom Reasoning 2, 287–327 (1986). https://doi.org/10.1007/BF02328452

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  • DOI: https://doi.org/10.1007/BF02328452

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