David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
Learn more about PhilPapers
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.
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
Thomas Jech, Set Theory. Journal of Symbolic Logic.
W. W. Tait (1981). Finitism. Journal of Philosophy 78 (9):524-546.
Penelope Maddy (1992). Indispensability and Practice. Journal of Philosophy 89 (6):275-289.
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.
Sylvia Wenmackers (2013). Ultralarge Lotteries: Analyzing the Lottery Paradox Using Non-Standard Analysis. Journal of Applied Logic 11 (4):452-467.
Similar books and articles
Yaroslav D. Sergeyev (2008). A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities. Informatica 19 (4):567-596.
Roman Duda (1997). Mathematics: Essential Tensions. [REVIEW] Foundations of Science 2 (1):11-19.
Ross Willard (1994). Hereditary Undecidability of Some Theories of Finite Structures. Journal of Symbolic Logic 59 (4):1254-1262.
Klaus Sutner (1990). The Ordertype of Β-R.E. Sets. Journal of Symbolic Logic 55 (2):573-576.
Pierluigi Miraglia (2000). Finite Mathematics and the Justification of the Axiom of Choicet. Philosophia Mathematica 8 (1):9-25.
Richard Heck (1998). The Finite and the Infinite in Frege's Grundgesetze der Arithmetik. In M. Schirn (ed.), Philosophy of Mathematics Today. OUP
Jan Mycielski (1981). Analysis Without Actual Infinity. Journal of Symbolic Logic 46 (3):625-633.
Herman Dishkant (1986). About Finite Predicate Logic. Studia Logica 45 (4):405 - 414.
Added to index2009-01-28
Total downloads39 ( #105,429 of 1,907,063 )
Recent downloads (6 months)3 ( #277,075 of 1,907,063 )
How can I increase my downloads?