Synthese 103 (3):389 - 420 (1995)

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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DOI 10.1007/BF01089734
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References found in this work BETA

Indispensability and Practice.Penelope Maddy - 1992 - Journal of Philosophy 89 (6):275.
Set Theory.Thomas Jech - 1981 - Journal of Symbolic Logic.
Finitism.W. W. Tait - 1981 - Journal of Philosophy 78 (9):524-546.
Understanding the Infinite.Shaughan Lavine - 1994 - Harvard University Press.

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Fair Infinite Lotteries.Sylvia Wenmackers & Leon Horsten - 2013 - Synthese 190 (1):37-61.

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