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Using a First Order Logic to Verify That Some Set of Reals Has No Lesbegue Measure

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Interactive Theorem Proving (ITP 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6172))

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Abstract

This paper presents a formal proof of Vitali’s theorem that not all sets of real numbers can have a Lebesgue measure, where the notion of “measure” is given very general and reasonable constraints. A careful examination of Vitali’s proof identifies a set of axioms that are sufficient to prove Vitali’s theorem, including a first-order theory of the reals as a complete, ordered field, “enough” sets of reals, and the axiom of choice. The main contribution of this paper is a positive demonstration that the axioms and inference rules in ACL2(r), a variant of ACL2 with support for nonstandard analysis, are sufficient to carry out this proof.

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Cowles, J., Gamboa, R. (2010). Using a First Order Logic to Verify That Some Set of Reals Has No Lesbegue Measure. In: Kaufmann, M., Paulson, L.C. (eds) Interactive Theorem Proving. ITP 2010. Lecture Notes in Computer Science, vol 6172. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14052-5_4

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  • DOI: https://doi.org/10.1007/978-3-642-14052-5_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14051-8

  • Online ISBN: 978-3-642-14052-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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