Finite mathematics
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.DOI
10.1007/bf01089734
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Citations of this work
Ultralarge lotteries: Analyzing the Lottery Paradox using non-standard analysis.Sylvia Wenmackers - 2013 - Journal of Applied Logic 11 (4):452-467.
References found in this work
Understanding the Infinite.Shaughan Lavine - 1994 - Cambridge, MA and London: Harvard University Press.