Monatshefte für Mathematik

, Volume 123, Issue 4, pp 309–319 | Cite as

Kinna-Wagner selection principles, axioms of choice and multiple choice

  • Paul Howard
  • Arthur L. Rubin
  • Jean E. Rubin
Article

Abstract

We study the relationships between weakened forms of the Kinna-Wagner Selection Principle (KW), the Axiom of Choice (AC), and the Axiom of Multiple Choice (MC).

1991 Mathematics Subject Classification

03E25 03E35 04A25 

Key words

Axiom of choice multiple choice Kinna-Wagner selection principle set theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Blass, A.: Ramsey's theorem in the hierarchy of choice principles. J. Symbolic Logic42, 387–390 (1977).Google Scholar
  2. [2]
    Blass, A.: Injectivity, projectivity and the axiom of choice. Trans. Amer. Math. Soc.255, 31–59 (1979).Google Scholar
  3. [3]
    Bleicher, M.: Multiple choice axioms and the axiom of choice for finite sets. Fund. Math.57, 247–252 (1965).Google Scholar
  4. [4]
    Brunner, N.: Dedekind-Endlichkeit und Wohlordenbarkeit. Mh. Math.94, 9–31 (1982).Google Scholar
  5. [5]
    Brunner, N.: Positive functionals and the axiom of choice. Rend. Sem. Mat. Univ. Padova72, 9–12 (1984).Google Scholar
  6. [6]
    Brunner, N.: The Fraenkel-Mostowski method, revisited. Notre Dame J. Formal Logic31, 64–75 (1990).Google Scholar
  7. [7]
    Cohen, P. J.: Set Theory and the Continuum Hypothesis. New York: Benjamin. 1966.Google Scholar
  8. [8]
    Feferman, S.: Applications of forcing and generic sets. Fund. Math.56, 325–345 (1965).Google Scholar
  9. [9]
    Felgner, U., Jech, T.: Variants of the axiom of choice in set theory with atoms. Fund. Math.79, 79–85 (1973).Google Scholar
  10. [10]
    Halpern, J., Levy, A.: The Boolean prime ideal theorem does not imply the axiom of choice. In: Axiomatic Set Theory. Proc. Symp. Pure Math (D. Scott ed.)13, 83–134 (1971).Google Scholar
  11. [11]
    Howard, P.: Limitations of the Fraenkel-Mostowski method of independence proofs. J. Symbolic Logic38, 416–422 (1973).Google Scholar
  12. [12]
    Howard, P.: Binary consistent choice on pairs and a generalization of König's infinity lemma. Fund. Math.121, 17–31 (1984).Google Scholar
  13. [13]
    Howard, P., Rubin, H., Rubin, J.: The relationship between two weak forms of the axiom of choice. Fund. Math.80, 75–79 (1973).Google Scholar
  14. [14]
    Howard P., Rubin, J.: The axiom of choice for well ordered families and for families of well orderable sets. J. Symbolic Logic.60, 1115–1117 (1995).Google Scholar
  15. [15]
    Jech, T.: The Axiom of Choice. Amsterdam: North Holland. 1973.Google Scholar
  16. [16]
    Jech, T., Sochor, A.: Applications of the Θ-model. Bull. de l'Acad. Pol. Sci.14, 297–303, 351–355 (1966).Google Scholar
  17. [17]
    Keremedis, K.: Bases for vector spaces over the two element field and the axiom of choice. Proc. Amer. Math. Soc.124, 2527–2531 (1996).Google Scholar
  18. [18]
    Keremedis, K.: Disasters in topology without the axiom of choice. Preprint (1995).Google Scholar
  19. [19]
    Kinna, W., Wagner, K.: Über eine Abschwächung des Auswahlpostulates. Fund. Math.42, 75–82 (1955).Google Scholar
  20. [20]
    Läuchli, H.: The independence of the ordering principle from a restricted axiom of choice. Fund. Math.54, 31–43 (1964).Google Scholar
  21. [21]
    Levy, A.: Axioms of multiple choice. Fund. Math.50, 475–483 (1962).Google Scholar
  22. [22]
    Mostowski, A.: Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip. Fund. Math.32, 201–252 (1939).Google Scholar
  23. [23]
    Pincus, D.: Support structures for the axiom of choice. J. Symbolic Logic36, 28–38 (1971).Google Scholar
  24. [24]
    Pincus, D.: Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods. J. Symbolic Logic37, 721–743 (1972).Google Scholar
  25. [25]
    Pincus, D.: Cardinal representatives. Israel J. Math. Logic18, 321–344 (1974).Google Scholar
  26. [26]
    Pincus, D.: Adding dependent choice. Ann. Math. Logic11, 105–145 (1977).Google Scholar
  27. [27]
    Pincus, D.: A note on the cardinal factorial. Fund. Math.98, 21–24 (1978).Google Scholar
  28. [28]
    Rubin, H.: Two propositions equivalent to the axiom of choice only under the axioms of extensionality and regularity. Notices7, 381 (1960).Google Scholar
  29. [29]
    Sageev, G.: An independence result concerning the axiom of choice. Ann. Math. Logic8, 1–184 (1975).Google Scholar
  30. [30]
    Truss, J.: The axiom of choice for linearly ordered families. Fund. Math.99, 133–139 (1978).Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Paul Howard
    • 1
  • Arthur L. Rubin
    • 2
  • Jean E. Rubin
    • 3
  1. 1.Department of MathematicsEastern Michigan UniversityYpsilonti
  2. 2.PhoenixUSA
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations