Abstract
I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals.
Similar content being viewed by others
REFERENCES
Balaguer, M.: 1998, ‘Non-Uniqueness as a Non-Problem’, Philosophia Mathematica (3) 6, 63–84.
Benacerraf, P.: 1965, ‘What Numbers Could Not Be’, in P. Benacerraf and H. Putnam (eds) (1984) Philosophy of Mathematics, Cambridge University Press, New York, pp. 272-295.
Benacerraf, P.: 1996, ‘What Mathematical Truth Could Not Be — I’, in A. Morton and S. Stich (eds), Benacerraf and his Critics, Blackwell, Cambridge, MA, pp. 9–59.
Boolos, G. and R. Jeffrey: 1974, Computability and Logic, Cambridge University Press, New York.
Church, A.: 1941, The Calculi of Lambda-Conversion, Annals of Mathematics Studies (6), Princeton University Press, Princeton, NJ.
Ciesielski, K.: 1997, Set Theory for the Working Mathematician, Cambridge University Press, New York.
Devlin, K: 1991, The Joy of Sets: Fundamentals of Contemporary Set Theory, Springer-Verlag, New York.
Drake, F.: 1974, Set Theory: An Introduction to Large Cardinals, American Elsevier, New York.
Eisenberg, M.: 1971, Axiomatic Theory of Sets and Classes, Holt, Rinehart, and Winston Inc., New York.
Enderton, H.: 1972, A Mathematical Introduction to Logic, Academic Press, New York.
Enderton, H.: 1977, Elements of Set Theory, Academic Press, New York.
Halmos, P.: 1960, Naive Set Theory, Van Nostrand, New York.
Hamilton, A.: 1982, Numbers, Sets, and Axioms: The Apparatus of Mathematics,Cambridge University Press, New York.
Hrbacek, K. and T. Jech: 1978, Introduction to Set Theory, Marcel Dekker Inc., New York.
Just, W. and M. Weese: 1996, Discovering Modern Set Theory I: The Basics, American Mathematical Society, Providence, RI.
Katz, J. J.: 1996, ‘Skepticism about Numbers and Indeterminacy Arguments’, in A. Morton and S. Stich (eds), Benacerraf and his Critics, Blackwell, Cambridge, MA, pp. 119–142.
Kessler, G.: 1980, ‘Frege, Mill, and the Foundations of Arithmetic’, Journal of Philosophy 77(2), 65–79.
Korner, S.: 1968, The Philosophy of Mathematics: An Introductory Essay, Dover Publications, New York.
Krivine, J.-L.: 1971, Introduction to Axiomatic Set Theory, D. Reidel Publishing, Dordrecht.
Maddy, P.: 1981, ‘Sets and Numbers’, Nous 15(4), 495–512.
Maddy, P.: 1992, Realism in Mathematics, Oxford University Press, New York.
Parsons, C.: 1994, ‘Intuition and Number’, in A. George (ed.), Mathematics and the Mind, Oxford University Press, New York, pp. 141–157.
Pinter, C.: 1971, Set Theory, Addison-Wesley, Reading, MA.
Quine, W. V. O.: 1970, Set Theory and Its Logic, Harvard University Press, Cambridge, MA.
Resnik, M.: 1997, Mathematics as a Science of Patterns, Oxford University Press, New York.
Shapiro, S.: 1997, Philosophy of Mathematics: Structure and Ontology, Oxford University Press, New York.
Suppes, P.: 1972, Axiomatic Set Theory, Dover, New York.
Wetzel, L.: 1989, ‘That Numbers Could be Objects’, Philosophical Studies 56, 273–292.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Steinhart, E. Why Numbers Are Sets. Synthese 133, 343–361 (2002). https://doi.org/10.1023/A:1021266519848
Issue Date:
DOI: https://doi.org/10.1023/A:1021266519848