Skip to main content
SpringerLink
Log in

The Inconsistency of Dedekind’s Infinite Set

  • Chapter
  • First Online:
Proofs of the Cantor-Bernstein Theorem

Part of the book series: Science Networks. Historical Studies ((SNHS,volume 45))

  • 1449 Accesses

Abstract

We review here the arguments raised by Cantor against Dedekind’s infinite set and Dedekind’s own doubts on this issue. This chapter still touches the main subject of this book, CBT, on two points: inconsistent sets and Bernstein’s visit to Dedekind which brought about Dedekind’s proof of CBT (see Chap. 4, Sect. 7.4, Chap. 9).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Contrary to this explicit timing, Mostowski, in Lakatos 1967 p 82, who quoted the last paragraph from Becker 1954 p 316, added that the said set descriptions were made in a conversation between Cantor and Dedekind. This additional information is not in Becker and seems to be a mistake.

  2. 2.

    Ewald has here “staring into the indeterminate”.

  3. 3.

    The official publication date of Zahlen is 1888 but in Dedekind’s unpublished paper (Sinaceur 1971 p 251) Dedekind remarks that Zahlen appeared on Christmas 1887.

  4. 4.

    We slightly changed Cantor’s notation in this paragraph.

  5. 5.

    The ambiguity whether a is a class or a power appears in the original.

  6. 6.

    ΣM a denoting the union of all M a .

  7. 7.

    Scharlau 1981 p 10 says that the meeting took place in Harzburg but Dedekind’s letter of August 29 is from Braunschweig and Dugac 1976 p 130 cites a postcard from Cantor dated September 3 announcing his coming to Braunschweig the next day.

  8. 8.

    Ebbinghaus (2007 p 85) calls Dedekind’s infinite set ‘inconsistent’ not in Cantor’s sense but to convey its contradictory character. In this he is not justified (see Fraenkel et al. 1973 p 86 footnote 4).

  9. 9.

    http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind1.html.

References

  • Becker O. Grundlagen der Mathematik in geschichtlicher Entwickelung, München; 1954.

    Google Scholar 

  • Bolzano B. Paradoxien des Unendlichen. Vdm Verlag, 2006. Partial translation in Ewald 1996 vol 1 p. 249.

    Google Scholar 

  • Cantor G. Grundlagen einer allegmeinen Mannigfaltigkeitslehre, (‘Grundlagen’). Cantor 1932;165–209. English translation: Ewald 1996 vol 2, 881–920. Cf. Parpart 1976.

    Google Scholar 

  • Cantor G. Über eine elementare Frage der Mannigfaltigkeitslehre, (‘1892 Frage’). Jahresbericht der Deutschen Mathematiker-Vereinigung. 1892;1:75–8. Cantor 1932;278–80. English translation Ewald 1996 vol 2, 920–2.

    Google Scholar 

  • Cantor G. Beiträge zur Begründung der transfiniten Mengenlehre, (‘1895 Beiträge’). Cantor 1932;282–311. English translation: Cantor 1915.

    Google Scholar 

  • Cantor G. Beiträge zur Begründung der transfiniten Mengenlehre, (‘1897 Beiträge’). Cantor 1932;312–56. English translation: Cantor 1915.

    Google Scholar 

  • Cantor G. Gesammelte Abhandlungen Mathematischen und philosophischen Inhalts, edited by Zermelo E. Springer, Berlin 1932. http://infini.philosophons.com/.

  • Cavailles J. Philosophie mathématique. Paris: Hermann; 1962.

    MATH  Google Scholar 

  • Charraud N. Infini et inconscient: essai sur Georg Cantor. Paris: Anthropos; 1994.

    MATH  Google Scholar 

  • Dedekind R. Gesammelte Mathematische Werke, edited by Fricke R, Noether E, Ore O. vol 3 Braunschweig, 1930–32.

    Google Scholar 

  • Dedekind R. Essays on the theory of numbers, English translation of Dedekind 1888, by Berman WW, New York: Dover; 1963. Also in Ewald 1996 vol 2 p 787–833.

    Google Scholar 

  • Dugac P. Richard Dedekind et les fondements des mathématiques. Paris: Vrin; 1976.

    MATH  Google Scholar 

  • Ebbinghaus HD. Ernst Zermelo. An approach to his life and work. New York: Springer; 2007.

    MATH  Google Scholar 

  • Ewald W. editor. From Kant to Hilbert: a source book in the foundations of mathematics. 2 vols. Oxford: Clarendon Press; 1996.

    Google Scholar 

  • Ferreirós J. Labyrinth of thought. A history of set theory and its role in modern mathematics. Basel/Boston/Berlin: Birkhäuser; 1999.

    MATH  Google Scholar 

  • Fraenkel AA. Abstract set theory. 3rd ed. Amsterdam: North Holland; 1966.

    Google Scholar 

  • Fraenkel AA, Bar-Hillel Y, Levy A. Foundations of set theory. 2nd ed. Amsterdam: North Holland; 1973.

    MATH  Google Scholar 

  • Frege G. Grundgesetze der Arithmetik, Begriffsscriftlich abgeleitet. 2 Bände. Jena. English translation: Montgomery Furth, University of California Press, Berkeley, 1967.

    Google Scholar 

  • Frege G. Kritische Beleuchtung einiger Punkte in E. Schröder’s Vorlesungen über die Algebra der Logik, Archiv fur systematische Philosophie 1, 433–56. English translation: Geach P. Black M. Basil, Translation from the philosophical writings of Gottlob Frege, Oxford: Blackwell; 1980. pp. 86–106.

    Google Scholar 

  • Gillies DA. Frege, Dedekind, and Peano on the foundations of arithmetic. Assen, Netherlands: Van Gorcum; 1982.

    MATH  Google Scholar 

  • Grattan-Guinness I. Towards a biography of Georg Cantor. Ann Sci. 1970;27:345–91.

    Google Scholar 

  • Hallett M. Cantorian set theory and limitation of size. Oxford: Clarendon Press; 1984.

    Google Scholar 

  • Hessenberg G. Grundbegriffe der Mengenlehre. Abhandlungen der Friesschen Schule. 1906;2(1):479–706. reprinted Göttingen, Vardenhoeck & Ruprecht 1906.

    Google Scholar 

  • Lakatos I. Proofs and refutations. Cambridge: Cambridge University Press; 1976.

    Book  MATH  Google Scholar 

  • Landau E. Richard Dedekind – Gedächtisrede, Nachrichten von der Koniglichen Gesellschaft der Wissenschaften zu Gottingen Geschäftliche Mitteilungen 1917;50–70.

    Google Scholar 

  • McCarty DC. The mysteries of Richard Dedekind. In: Hintikka J, editor. From Dedekind to Gödel. Dordrecht: Kluwer; 1995. p. 53–96.

    Chapter  Google Scholar 

  • Moore GH, Garciadiego A. Burali-Forti’s Paradox: a reappraisal of its origins. Hist Math. 1981;8:319–50.

    Article  MathSciNet  MATH  Google Scholar 

  • Pla i Carrera J. Dedekind and the theory of sets. Modern Logic. 1993;3(3):215–305.

    MathSciNet  MATH  Google Scholar 

  • Purkert W, Ilgauds HJ. Georg Cantor 1845–1918. Basel: Birkhäuser Verlag; 1987.

    MATH  Google Scholar 

  • Russell B. Introduction to mathematical philosophy. London: George Allen & Unwin; 1919.

    MATH  Google Scholar 

  • Russell B. The collected papers of Bertrand Russell: foundations of logic, 1903–1905, vol 4, edited by Urquart A, London: Routledge; 1994.

    Google Scholar 

  • Scharlau W. editor. Richard Dedekind 1831–1981. Eine Würdigung zu seinem 150 Geburtstag, Braunschweig/Weisbaden: Vieweg; 1981.

    Google Scholar 

  • Sinaceur MA. Appartenance et inclusion: un inedit de Richard Dedekind. Revue d’Histoire des Sciences. 1971;24:247–55.

    Article  MATH  Google Scholar 

  • Zermelo E. Neuer Beweiss für die Möglichkeit einer wohlordnung, Mathematische Annalen 1908a;65:107–28. English translation: van Heijenoort 1967;183–98.

    Google Scholar 

  • Zermelo E. Untersuchungen über die Grundlagen der Mengenlehre. I, Mathematische Annalen 1908b;65:261–81. English translation: van Heijenoort 1967;199–215.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Cohn Institute for the History and Philosophy of Science and Ideas, Tel Aviv University, Tel Aviv, Israel

    Arie Hinkis

Authors
  1. Arie Hinkis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arie Hinkis .

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Basel

About this chapter

Cite this chapter

Hinkis, A. (2013). The Inconsistency of Dedekind’s Infinite Set. 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_8

Download citation

Publish with us

Policies and ethics

Search

Navigation