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
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