Hostname: page-component-7dd5485656-wjfd9 Total loading time: 0 Render date: 2025-10-31T05:20:08.270Z Has data issue: false hasContentIssue false
  • English
  • Français

On Ackermann's set theory1

Published online by Cambridge University Press:  12 March 2014

Azriel Lévy
Azriel Lévy*
Affiliation:
Massachusetts Institute of Technology, Cambridge and Hebrew University, Jerusalem
Get access Rights & Permissions [Opens in a new window]

Extract

Ackermann introduced in [1] a system of axiomatic set theory. The quantifiers of this set theory range over a universe of objects which we call classes. Among the classes we distinguish the sets. Here we shall show that, in some sense, all the theorems of Ackermann's set theory can be proved in Zermelo-Fraenkel's set theory. We shall also show that, on the other hand, it is possible to prove in Ackermann's set theory very strong theorems of the Zermelo-Fraenkel set theory.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 24 , Issue 2 , June 1959 , pp. 154 - 166
Copyright
Copyright © Association for Symbolic Logic 1959

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.)

Article purchase

Temporarily unavailable

Footnotes

1

The ideas on which this paper is based are contained in the author's Ph.D. thesis submitted to the Hebrew University. The author wishes to express his gratitude to Prof. A. A. Fraenkel and Prof. A. Robinson for their guidance and kind encouragement. This paper was written while the author was a Sloan Fellow of the School for Advanced Study at the Massachusetts Institute of Technology.

References

[1]Ackermann, W., Zur Axiomatik der Mengenlehre, Mathematische Annalen, vol. 131 (1956), pp. 336–345.CrossRefGoogle Scholar
[2]Fraenkel, A. A., Einleitung in die Mengenlehre, 3rd edition, Berlin 1928.CrossRefGoogle Scholar
[3]Fraenkel, A. A. and Bar-Hillel, Y., Foundations of set theory, Amsterdam 1958.Google Scholar
[4]Gödel, K., The consistency of the continuum hypothesis, Princeton 1953.Google Scholar
[5]Lévy, A., Axiom schemata of strong infinity in axiomatic set theory, to appear in the Pacific journal of mathematics.Google Scholar
[6]Montague, R. and Feferman, S., The method of arithmetization and some of its applications, North-Holland Pub. Co., to appear.Google Scholar
[7]von Neumann, J., Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre, Mathematische Annalen, vol. 99 (1928), pp. 373–391.CrossRefGoogle Scholar
[8]Shepherdson, J. C., Inner models for set theory, Part II, this Journal, vol. 17 (1952), pp. 225–237.Google Scholar
[9]Tarski, A., Mostowski, A., and Robinson, R. M., Undecidable theories, Amsterdam 1953.Google Scholar
[10]Tarski, A. and Vaught, R. L., Arithmetical extensions of relational systems, Compositio mathematica, vol. 13 (1957), pp. 81–102.Google Scholar
[11]Church, A., Introduction to mathematical logic, Princeton, 1956.Google Scholar