Abstract
In this and the next chapter we present the complex of ideas surrounding Cantor’s theory of inconsistent sets. Our discussion serves to stress the possibility and importance of reading Cantor in his own context. For Cantor the scale of numbers and number-classes was the backbone of his set theory and the theory of inconsistent sets its necessary frame. It is unfortunately common to take towards Cantor’s work the Zermelo-Dedekind approach, which by-passes these notions, up to and including the solution of the comparability of sets through the Well-Ordering Theorem. Such approach goes against the way Cantor had developed his theory, both conceptually and historically.
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Notes
- 1.
In the October 2, 1897, letter to Hilbert (Ewald 1996 vol 2 p 927f) the requirement is that the set be “finished”.
- 2.
Cantor did not use systematically this adjective to qualify the not inconsistent sets.
- 3.
- 4.
See Levy’s proof (1979 p 11) that Russell’s set is inconsistent.
- 5.
- 6.
To be distinguished from Ω of Grundlagen.
- 7.
Hence, a set that contains an inconsistent set is inconsistent.
- 8.
Cantor did not have a rule that corresponds to the power-set axiom, say, that the set of functions from a set to a set is a set.
- 9.
- 10.
- 11.
Cantor uses this Dedekindian term ‘system’ and other terms (e.g., ‘multiplicity’ above) to avoid referring to inconsistent sets by the term ‘set’.
- 12.
In the attachment for Schoenflies, to his letter of June 28, 1899, to Hilbert (Meschkowski-Nilson 1991 p 403), Cantor noted without any argument that Theorem A follows from the thesis that all cardinal numbers are alephs, which in turn follows from the fact that the collection of all alephs is inconsistent (‘not finished’ in the terminology of that letter). Schoenflies did nothing with this remark which is indeed quite obscure. It can be deciphered only by reference to the letter to Dedekind.
- 13.
This procedure was indeed used by Jourdain in his proof of a similar result (1904a p 70, cf. p 67; see Chap. 17).
- 14.
The context of Corollary D will be discussed in the next chapter. There we will explain our reasons for believing that Cantor intended to postulate Corollary D.
- 15.
Note that in Zermelo’s axiomatic set theory the axiom of choice applies to consistent sets only. In the next chapter we will show how the Comparability Theorem for cardinal numbers follows from Theorem C above. Thus Cantor obtained the Well-Ordering Theorem for consistent sets. But with Corollary D as postulate, Cantor also obtained the comparability of all sets, namely, also that of the inconsistent sets.
- 16.
- 17.
This was Zermelo’s aim, which he achieved by limiting his theory to the members of a certain model of his axioms. The members of the model are all consistent sets. The inconsistent sets are the classes in Zermelo’s theory, and about them Zermelo did not prove even CBT.
- 18.
- 19.
- 20.
- 21.
The theorem uses infinite choices, as was already noted by Bettazzi (1896 p 512 footnote (1), cf. Moore 1982 p 30, Ferreirós 1999 p 313), who thought that the use of such an axiom is “inappropriate”. Bettazzi was a member of the group of mathematicians associated with Peano. The latter was the first to spot and object the use of infinite choices (1890). Cf. Chap. 20
- 22.
- 23.
- 24.
- 25.
It is possible that what Cantor found in October 1882 (see the letter to Dedekind cited in Sect. 2.3) was the idea of inconsistent sets and that the other ideas he had obtained earlier.
- 26.
Note that even after this move, the principles of generation of 1883 Grundlagen are necessary to generate new well-ordered sets, in the sequence of number-classes.
References
Aczel AD. The mystery of the aleph. New York: Washington Square Press; 2000.
Bettazzi R. Gruppi finiti ed infiniti di enti, Atti della Reale Academia delle scienze di Torino, Classe di scienze fisiche, matematiche e naturali, 1896;31:506–12. http://www.archive.org/details/attidellarealeac31real.
Bolzano B. Paradoxien des Unendlichen. Vdm Verlag, 2006. Partial translation in Ewald 1996 vol 1 p. 249.
Bunn R. Developments in the foundation of mathematics from 1870 to 1910. In: Grattan-Guinness I, editor. From the calculus to set theory, 1630–1910: an introductory history. Princeton: Princeton University Press; 1980.
Cantor G. Ein Beitrag zur Mannigfeltigkeitslehre, (‘1878 Beitrag’). Cantor 1932;119–33.
Cantor G. Beiträge zur Begründung der transfiniten Mengenlehre, (‘1895 Beiträge’). Cantor 1932;282–311. English translation: Cantor 1915.
Cantor G. Gesammelte Abhandlungen Mathematischen und philosophischen Inhalts, edited by Zermelo E. Springer, Berlin 1932. http://infini.philosophons.com/.
Cantor G. Principien einer Theorie der Ordnungstypen, erste Mitteilung, in Grattan-Guinness 1970.
Cavailles J. Philosophie mathématique. Paris: Hermann; 1962.
Charraud N. Infini et inconscient: essai sur Georg Cantor. Paris: Anthropos; 1994.
Dauben JW. Georg Cantor. His Mathematics and the Philosophy of the Infinite, Cambridge MA: Harvard University Press; 1979. Reprinted by Princeton University Press, 1990.
Dauben JW. Georg Cantor and the battle for transfinite set theory, http://www.acmsonline.org/Dauben-Cantor.pdf.
Dedekind R. Gesammelte Mathematische Werke, edited by Fricke R, Noether E, Ore O. vol 3 Braunschweig, 1930–32.
Dugac P. Richard Dedekind et les fondements des mathématiques. Paris: Vrin; 1976.
Ewald W. editor. From Kant to Hilbert: a source book in the foundations of mathematics. 2 vols. Oxford: Clarendon Press; 1996.
Ferreirós J. Labyrinth of thought. A history of set theory and its role in modern mathematics. Basel/Boston/Berlin: Birkhäuser; 1999.
Frege G. Grundlagen der Arithmetik, Breslau: Verlag von Wilhelm Koebner. English translation: J.L. Austin, Oxford, Basil Blackwell, 1950.
Grattan-Guinness I. An unpublished paper of Georg Cantor. Acta Math. 1970;124:65–107.
Grattan-Guinness I. The correspondence between Georg Cantor and Philip Jourdain. Jahresbericht der Deutschen Mathematiker-Vereinigung. 1971a;73(Part 1):111–39.
Grattan-Guinness I. The rediscovery of the Cantor-Dedekind correspondence. Jahresbericht der Deutschen Mathematiker-Vereiningung. 1974;76:104–39.
Grattan-Guinness I. Dear Russell - dear Jourdain. A commentary on Russell’s logic, based on his correspondence with Philip Jourdain. New York: Columbia University Press; 1977.
Hallett M. Cantorian set theory and limitation of size. Oxford: Clarendon Press; 1984.
van Heijenoort J. From Frege to Gödel. Cambridge, MA: Harvard University Press; 1967.
Hilbert D. Über die Grundlagen der Logik und Arithmetic, Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904, 174–185. English translation: van Heijenhoort 1967. pp. 129–38.
Jané I. The role of the absolute infinite in Cantor’s conception of set. Erkenntnis 1905;42:375–402.
Levy A. Basic set theory, Dover Publications Inc 2002. Originally published by Springer in 1979.
Meschkowski H, Nilsen W. Georg Cantor: briefe. Berlin: Springer; 1991.
Moore GH. The origins of Zermelo’s axiomatisation of set theory. J Philosophical Logic. 1978;7:307–29.
Moore GH. Zermelo’s axiom of choice: its origin, development and influence. Berlin: Springer; 1982.
Purkert W, Ilgauds HJ. Georg Cantor 1845–1918. Basel: Birkhäuser Verlag; 1987.
Tait WW. Cantor’s Grundlagen and the Paradoxes of set theory. http://home.uchicago.edu/~wwtx/cantor.pdf.
Zermelo E. Untersuchungen über die Grundlagen der Mengenlehre. I, Mathematische Annalen 1908b;65:261–81. English translation: van Heijenoort 1967;199–215.
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Hinkis, A. (2013). The Theory of Inconsistent Sets. In: Proofs of the Cantor-Bernstein Theorem. Science Networks. Historical Studies, vol 45. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0224-6_4
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