Abstract
In 1887 Georg Cantor gave an influential but cryptic proof of theimpossibility of infinitesimals. I first give a reconstruction ofCantor's argument which relies mainly on traditional assumptions fromEuclidean geometry, together with elementary results of Cantor's ownset theory. I then apply the reconstructed argument to theinfinitesimals of Abraham Robinson's nonstandard analysis. Thisbrings out the importance for the argument of an assumption I call theChain Thesis. Doubts about the Chain Thesis are seen to render thereconstructed argument inconclusive as an attack on the infinitelysmall.
Similar content being viewed by others
REFERENCES
Boolos, G. and R. Jeffrey: 1989, Computability and Logic, 3rd edn, Cambridge University Press, Cambridge.
Borsuk, K. and W. Szmielew: 1960, Foundations of Geometry, North-Holland, Amsterdam. Revised English translation.
Cantor, G.: 1887, ‘Mitteilungen zur Lehre vom Transfiniten’, Zeitschrift für Philosophie und philosopische Kritik 91, 81–125; (1888), 92, 240-265. Page references are to Cantor (1932).
Cantor, G.: 1932, ‘Mitteilungen zur Lehre vom Transfiniten’, in E. Zermelo (ed.), Gesammelte Abhandlungen, Georg Olms, Hildesheim, pp. 378–439.
Dauben, J. W.: 1990, Georg Cantor: His Mathematics and Philosophy of the Infinite, reprint edn, Princeton University Press, Princeton.
Ehrlich, P.: 1994, ‘General Introduction’, in P. Ehrlich (ed.), Real Numbers, Generalizations of the Reals, and Theories of Continua, Kluwer, Boston, pp. 7–32.
Ehrlich, P.: 1995, ‘Hahn's Ñber die Nichtarchimedischen Größensysteme and the Development of the Modern Theory of Magnitudes and Numbers to Measure Them’, in J. Hintikka (ed.), Essays on the Development of the Foundations of Mathematics, Kluwer, Boston, pp. 165–213.
Ehrlich, P.: 1997, ‘From Completeness to Archimedean Completeness’, Synthese 110, 57–76.
Goldblatt, R.: 1998, Lectures on the Hyperreals, Graduate Texts in Mathematics, Springer, New York.
Hartshorne, R.: 2000, Geometry: Euclid and Beyond, Springer, New York.
Henkin, L.: 1950, ‘Completeness in the Theory of Types’, Journal of Symbolic Logic 15, 81–91.
Hilbert, D.: 1899, Grundlagen der Geometrie, Teubner, Leipzig.
Hilbert, D.: 1968, Grundlagen der Geometrie, Teubner, Stuttgart, 10th edn of Hilbert (1899), with supplement by P. Bernays; page references are to Hilbert (1971).
Hilbert, D.: 1971, Foundations of Geometry, 2nd English edn, Open Court, La Salle, IL. Translation by Leo Unger of Hilbert (1968).
Peano, G.: 1892, ‘Dimostrazione dell'impossibilità di segmenti infinitesimi costanti’, Rivista di Matematica 2, 58–62.
Robinson, A.: 1979, ‘Non-Standard Analysis’, in H. Keisler, S. Körner, W. Luxemburg, and A. Young (eds.), Selected Papers, Vol. 2 (Nonstandard Analysis and Philosophy), Yale University Press, New Haven, pp. 3–11.
Russell, B.: 1903, Principles of Mathematics, Cambridge University Press, Cambridge. Page references are to Russell (1938).
Russell, B.: 1938, Principles of Mathematics, 2nd edn, Norton, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Moore, M.E. A Cantorian Argument Against Infinitesimals. Synthese 133, 305–330 (2002). https://doi.org/10.1023/A:1021204522829
Issue Date:
DOI: https://doi.org/10.1023/A:1021204522829