Abstract
This postscript to Part I consists of philosophical and historical remarks concerning the nature of the natural numbers. It contrasts the absolutism absolutist approach requiring absolute constructions of individual natural numbers such as those given by Frege, G.Frege, Russell, B.Russell, Zermelo, E.Zermelo, and von Neumann, with Dedekind, R.Dedekind’s structuralism structuralist approach in which the natural numbers can be taken as members of any Dedekind–Peano system.
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Notes
- 1.
Equinumerosity and cardinal numbers will be studied in Chap. 5.
- 2.
Problems arise with the naive Frege–Russell invariant (Chap. 20) which can only be addressed by using less natural approaches such as Quine’s New Foundations or Scott’s modified invariant (Definition 1297) in the context of ZF set theory.
- 3.
This existence result is related to the axiom of!infinity—nnnAxiom of Infinity and its equivalent forms. See the part of Sect. 21.5 dealing with the Axiom of Infinity where this topic is further discussed, particularly Theorem 1263, as well as Problem 1225.
- 4.
- 5.
Even in our absolute constructions of the previous chapters for extending the system of natural numbers to larger and larger systems of numbers such as the ratios, the lengths, the real numbers, and the complex numbers, we had already used the structuralist approach by throwing away old entities and replacing them with “isomorphic copies” found within the new extensions.
References
R. Dedekind. The nature and meaning of numbers (1888). In Essays on the Theory of Numbers [12]. English translation of “Was sind und was sollen die Zahlen?”, Vieweg, 1888.
R. Dedekind. Essays on the Theory of Numbers. Public Domain (originally published by Open Court), 1901.
G. Link, editor. One Hundred Years of Russell’s Paradox: Mathematics, Logic, Philosophy, volume 6 of de Gruyter Series in Logic and Its Applications. Walter de Gruyter, 2004.
E. H. Reck. Dedekind’s structuralism: An interpretation and partial defense. Synthese, 137(3):369–419, 2003.
B. Russell. Introduction to Mathematical Philosophy. Public Domain (originally published by George Allen & Unwin and Macmillan), 2nd edition, 1920.
S. Shapiro. Philosophy of Mathematics: Structure and Ontology. Oxford University Press, 1997.
S. Shapiro. The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press, 2005.
E. Zermelo. Investigations in the foundations of set theory I (1908). In From Frege to Gödel [78], pages 199–215. English translation of “Untersuchungen über die Grundlagen der Mengenlehre I”, Mathematische Annalen, 65(2):261–281, 1908.
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Dasgupta, A. (2014). Postscript I: What Exactly Are the Natural Numbers?. In: Set Theory. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-8854-5_4
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