Proofs of the Cantor-Bernstein Theorem pp 15-25 | Cite as
Generalizing Cantor’s CBT Proof
Chapter
First Online:
Abstract
Following his statement of CBT, in its single-set formulation, for sets of the power of (II), as a corollary to the Fundamental Theorem, Cantor said (Cantor 1932 p 201, Ewald 1996 vol 2 p 912 [12]):
Keywords
Limitation Theorem Induction Hypothesis Arithmetic Operation Union Theorem Fundamental Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
- Cantor G. Über unendliche, lineare Punktmannigfalitgkeiten, 6, Mathematische Annalen. 1884;23:453–88. Cantor 1932, 210–46.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. Contributions to the founding of the theory of Transfinite Numbers, English version of Cantor 1895 and Cantor 1897, translated by Jourdain PEB. Dover Publications Inc.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.MATHGoogle Scholar
- Dauben JW. Georg Cantor. His Mathematics and the Philosophy of the Infinite, Cambridge MA: Harvard University Press; 1979. Reprinted by Princeton University Press, 1990.Google Scholar
- Dugac P. Richard Dedekind et les fondements des mathématiques. Paris: Vrin; 1976.MATHGoogle Scholar
- Ewald W. editor. From Kant to Hilbert: a source book in the foundations of mathematics. 2 vols. Oxford: Clarendon Press; 1996.Google Scholar
- Felscher W. Did Cantor prove the Schröder-Bernstein theorem?, http://sunsite.utk.edu/math_archives/.http/hypermail/historia/mar99/0148.html.
- Ferreirós J. ‘What fermented in me for years’: Cantor’s discovery of transfinite numbers. Hist Math. 1995;22:33–42.MATHCrossRefGoogle Scholar
- Ferreirós J. Labyrinth of thought. A history of set theory and its role in modern mathematics. Basel/Boston/Berlin: Birkhäuser; 1999.MATHGoogle Scholar
- Fraenkel AA. Abstract set theory. 3rd ed. Amsterdam: North Holland; 1966.Google Scholar
- Grattan-Guinness I. The rediscovery of the Cantor-Dedekind correspondence. Jahresbericht der Deutschen Mathematiker-Vereiningung. 1974;76:104–39.MathSciNetMATHGoogle Scholar
- Grattan-Guinness I. The search for mathematical roots, 1870–1940: logics, set theories and the foundations of mathematics from Cantor through Russell and Gödel, Princeton University Press; 2000.Google Scholar
- Hallett M. Cantorian set theory and limitation of size. Oxford: Clarendon Press; 1984.Google Scholar
- Hausdorff F. Grundzuge der Mengenlehre, Berlin; 1914a, reprinted by Chelsea, New York; 1949.Google Scholar
- van Heijenoort J. From Frege to Gödel. Cambridge, MA: Harvard University Press; 1967.MATHGoogle Scholar
- Hessenberg G. Grundbegriffe der Mengenlehre. Abhandlungen der Friesschen Schule. 1906;2(1):479–706. reprinted Göttingen, Vardenhoeck & Ruprecht 1906.Google Scholar
- Jourdain PEB. On the transfinite cardinal numbers of number-classes in general. Philosophical Magazine (6). 1904b;7(39):294–303.MATHCrossRefGoogle Scholar
- Jourdain PEB. The multiplication of alephs. Mathematische Annalen. 1908a;65:506–12.MathSciNetMATHCrossRefGoogle Scholar
- Levy A. Basic set theory, Dover Publications Inc 2002. Originally published by Springer in 1979.Google Scholar
- Lindenbaum A, Tarski A. Communication sur les recherches de la thèorie des ensembles. Comptes rendu des séances de la société Polonaise de Mathematique section Varsovie Annales de la Societe Polonaise Mathematique. 1926;19:299–330.MATHGoogle Scholar
- Medvedev FA. 1966. Ранняя история теоремы эквивалентности (Early history of the equivalence theorem), Ист.-мат. исслед. (Research in the history of mathematics) 1966;17:229–46.Google Scholar
- Meschkowski H, Nilsen W. Georg Cantor: briefe. Berlin: Springer; 1991.Google Scholar
- Schröder E. Über Zwei Defitionen der Endlichkeit und G. Cantorsche Sätze, Nova Acta. Abhandlungen der Kaiserlichen Leopold-Carolinschen deutchen Akademie der Naturfoscher. 1898;71:301–62.Google Scholar
- Tait WW. Book review on Potter 2004. History and philosophy of logic. 2005;26(2):164.Google Scholar
Copyright information
© Springer Basel 2013