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On the effectiveness of the Schröder-Bernstein theorem


Author: J. B. Remmel
Journal: Proc. Amer. Math. Soc. 83 (1981), 379-386
MSC: Primary 03D20
DOI: https://doi.org/10.1090/S0002-9939-1981-0624936-X
MathSciNet review: 624936
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Abstract: The effectiveness of the classical equivalence theorem of Schröder and Bernstein is investigated using the tools of recursion theory. We prove one result which generalizes all the effective versions of the Schröder-Bernstein theorem which occur in the literature. In contrast, we show that Banach's strengthening of the Schröder-Bernstein theorem fails to be effective.


References [Enhancements On Off] (What's this?)

  • [1] S. Banach, Un théorème sur les transformations biunivoques, Fund. Math. 6 (1924), 236-239.
  • [2] E. Borel, Leçons sur la théorie des fonctions, Paris, 1898.
  • [3] G. Cantor, Beiträge zur Begründung der transfiniten Mengenlehre. II, Math. Ann. 49 (1897), 207-246. MR 1510964
  • [4] J. C. E. Dekker and J. Myhill, Recursive equivalence types (Monograph), Univ. California Publ. Math. 3 (1960). MR 0117155 (22:7938)
  • [5] A. Korselt, Über einen Beweis des Äquivalenzsatzes, Math. Ann. 70 (1911), 294-296. MR 1511622
  • [6] A. Manaster and J. Rosenstein, Effective matchmaking (recursion theoretic aspects of a theorem of Philip Hall), Proc. London Math. Soc. 25 (1972), 615-654. MR 0314610 (47:3161)
  • [7] L. Mirsky and H. Perfect, Systems of representatives, J. Math. Anal. Appl. 15 (1966), 520-568. MR 0204300 (34:4144)
  • [8] J. Myhill, Creative sets, Z. Math. Logik Grundlagen Math. 1 (1955), 97-108. MR 0071379 (17:118g)
  • [9] H. Rogers, Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967. MR 0224462 (37:61)
  • [10] J. Shoenfield, Degrees of unsolvability, North-Holland, New York, 1971. MR 0340011 (49:4768)

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DOI: https://doi.org/10.1090/S0002-9939-1981-0624936-X
Article copyright: © Copyright 1981 American Mathematical Society

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