David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Synthese 103 (3):389 - 420 (1995)
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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References found in this work BETA
Thomas Jech, Set Theory. Journal of Symbolic Logic.
Penelope Maddy (1992). Indispensability and Practice. Journal of Philosophy 89 (6):275-289.
W. W. Tait (1981). Finitism. Journal of Philosophy 78 (9):524-546.
Alfred Tarski & Steven R. Givant (1987). A Formalization of Set Theory Without Variables. Monograph Collection (Matt - Pseudo).
Shaughan Lavine (1994). Understanding the Infinite. Harvard University Press.
Citations of this work BETA
Sylvia Wenmackers & Leon Horsten (2013). Fair Infinite Lotteries. Synthese 190 (1):37-61.
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