Abstract
A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories.
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I would like to thank Jeff Barrett, Akeel Bilgrami, Leigh Cauman, John Collins, William Craig, Gary Feinberg, Haim Gaifman, Yair Guttmann, Hidé Ishiguro, Isaac Levi, James Lewis, Vann McGee, Sidney Morgenbesser, George Shiber, Sarah Stebbins, Mark Steiner, and an anonymous referee for encouragement and various useful suggestions. The research described in this article and the preparation of the article were supported in part by the Columbia University Council for Research in the Humanities.
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Lavine, S. Finite mathematics. Synthese 103, 389–420 (1995). https://doi.org/10.1007/BF01089734
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DOI: https://doi.org/10.1007/BF01089734