Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-23T22:05:30.724Z Has data issue: false hasContentIssue false
  • English
  • Français

Limitations on the Fraenkel-Mostowski method of independence proofs

Published online by Cambridge University Press:  12 March 2014

Paul E. Howard
Paul E. Howard*
Affiliation:
Eastern Michigan University, Ypsilanti, Michigan 48197
Get access Rights & Permissions [Opens in a new window]

Abstract

The Fraenkel-Mostowski method has been widely used to prove independence results among weak versions of the axiom of choice. In this paper it is shown that certain statements cannot be proved by this method. More specifically it is shown that in all Fraenkel-Mostowski models the following hold:

1. The axiom of choice for sets of finite sets implies the axiom of choice for sets of well-orderable sets.

2. The Boolean prime ideal theorem implies a weakened form of Sikorski's theorem.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 3 , September 1973 , pp. 416 - 422
Copyright
Copyright © Association for Symbolic Logic 1973

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Gauntt, R. J., Some restricted versions of the axiom of choice, Notices of the American Mathematical Society, vol. 15 (1968), p. 351.Google Scholar
[2]Gauntt, R. J., Axiom of choice for finite sets—a solution to a problem of Mostowski, Notices of the American Mathematical Society, vol. 17 (1970), p. 454.Google Scholar
[3]Halpern, J. D., The independence of the axiom of choice from the Boolean prime ideal theorum, Fundamenta Mathematicae, vol. 55 (1963), pp. 57–66.Google Scholar
[4]Lauchli, H., The independence of the ordering principle from a restricted axiom of choice, Fundamenta Mathematicae, vol. 54 (1963), pp. 31–43.CrossRefGoogle Scholar
[5]Levy, A., The Fraenkel-Mostowski method for independence proofs, International Symposium on the Theory of Models (University of California, Berkeley, 1963), North-Holland, Amsterdam, 1965.Google Scholar
[6]Mostowski, A., On the principle of dependent choices, Fundamenta Mathematicae, vol. 35 (1948), pp. 127–130.Google Scholar
[7]Mostowski, A., The axiom of choice for finite sets, Fundamenta Mathematicae, vol. 33 (1945), pp. 137–168.CrossRefGoogle Scholar
[8]Mostowski, A., Über die Unabhangigkeit des Wohlordnungssatz vom Ordnungsprinzip, Fundamenta Mathematicae, vol. 32 (1939), pp. 201–252.CrossRefGoogle Scholar
[9]Sikorski, Roman, Boolean algebras, 2nd ed., Springer-Verlag, Berlin, 1964.Google Scholar