Classical arithmetic is quite unnatural

Jean Paul Van Bendegem



It is a generally accepted idea that strict finitism is a rather marginal view within the community of philosophers of mathematics. If one therefore wants to defend such a position (as the present author does), then it is useful to search for as many different arguments as possible in support of strict finitism. Sometimes, as will be the case in this paper, the argument consists of, what one might call, a “rearrangement” of known materials. The novelty lies precisely in the rearrangement, hence on the formal-axiomatic level most of the results presented here are not new. In fact, the basic results are inspired by and based on Mycielski (1981).

Full Text:



V. Allis and Teun Koetsier, ‘On Some Paradoxes of the Infinite’, British Journal for the Philosophy of Science, vol. 42, 1991, pp. 187–194.

V. Allis and Teun Koetsier, ‘On Some Paradoxes of the Infinite II’, British Journal for the Philosophy of Science, vol. 46, 1995, pp. 235–247.

Diderik Batens, ‘A General Characterization of Adaptive Logics’, Logique et Analyse, 2002 (to appear).

Diderik Batens, ‘The Demise of Rich Finitism. A Study in the Limitations of Paraconsistency’, Unpublished paper (available from the url:

Richard L. Epstein and Walter A. Carnielli, Computability. Computable Functions, Logic, and the Foundations of Mathematics, London, Wadsworth, 2000 (2nd edition).

P. Holgate, ‘Discussion: Mathematical Notes on Ross’s Paradox’, British Journal for the Philosophy of Science, vol. 45, 1994, pp. 302–304.

Richard Kaye, Models of Peano Arithmetic, Oxford, Clarendon Press, 1991.

Christian Michaux, (ed.), Definability in Arithmetics and Computability, Louvain-la-Neuve, Academia Bruylant, 2000 (Cahiers du Centre de Logique 11).

Jan Mycielski, ‘Analysis without Actual Infinity’, Journal of Symbolic Logic, vol. 46, number 3, 1981, pp. 625–633.

José Perez Laraudogoitia, ‘Supertasks’, in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Summer 2002 Edition), archives/sum2002/entries/spacetime-supertasks/.

Edward Nelson, Predicative Arithmetic, Princeton, Princeton University Press, 1986.

Jean Paul Van Bendegem, ‘Ross’ Paradox is an Impossible Super Task’, British Journal for the Philosophy of Science, vol. 45, 1994a, pp. 743–48.

Jean Paul Van Bendegem, ‘Strict Finitism as a Viable Alternative in the Foundations of Mathematics’, Logique et Analyse, vol. 37, 145, 1994b (date of publication: 1996), pp. 23–40.

Jean Paul Van Bendegem, ‘Why the largest number imaginable is still a finite number’, Logique et Analyse, vol. 41, 161–162–163, 1998 (date of publication: 2001), pp. 107–126.

Timothy Vermeir, ‘Inconsistency Adaptive Arithmetic’, Logique et Analyse, vol. 42, 167-168, 1999 (date of publication: 2002), pp. 221–241.

Ludwig Wittgenstein, Remarks on the Foundations of Mathematics, (Edited by G.H. von Wright, R. Rhees, G.E.M. Anscombe, translated by G.E.M. Anscombe), Oxford, Basil Blackwell, 19561, 19672, 19783 (revised and reset).

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

Partnerzy platformy czasopism