Hostname: page-component-784d4fb959-xqrh7 Total loading time: 0 Render date: 2025-07-14T17:51:26.069Z Has data issue: false hasContentIssue false
  • English
  • Français

Elementary embeddings and infinitary combinatorics

Published online by Cambridge University Press:  12 March 2014

Kenneth Kunen
Kenneth Kunen*
Affiliation:
University of Wisconsin, Madison, Wisconsin
Get access Rights & Permissions [Opens in a new window]

Extract

One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings, j, from the universe, V, into some transitive submodel, M. See Reinhardt–Solovay [7] for more details. If j is not the identity, and κ is the first ordinal moved by j, then κ is a measurable cardinal. Conversely, Scott [8] showed that whenever κ is measurable, there is such j and M. If we had assumed, in addition, that , then κ would be the κth measurable cardinal; in general, the wider we assume M to be, the larger κ must be.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 3 , September 1971 , pp. 407 - 413
Copyright
Copyright © Association for Symbolic Logic 1972

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

References

[1]Erdös, P. and Hajnal, A., On a problem of B. Jónsson, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 14 (1966), pp. 19–23.Google Scholar
[2]Gaifman, H., Pushing up the measurable cardinal, Lecture notes 1967 Institute on Axiomatic Set Theory (University of California, Los Angeles, Calif., 1967), American Mathematical Society, Providence, R.I., 1967, pp. IV R 1–16. (Mimeographed).Google Scholar
[3]Gödel, K., The consistency of the continuum hypothesis, Princeton Univ. Press, Princeton, 1940.CrossRefGoogle Scholar
[4]Kelley, J. L., General topology, D. Van Nostrand Co., Inc., Princeton, 1965.Google Scholar
[5]Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179–227.CrossRefGoogle Scholar
[6]Kunen, K., On the GCH at measurable cardinals, Proceedings of 1969 Logic Summer School at Manchester, pp. 107–110.Google Scholar
[7]Reinhardt, W. and Solovay, R., Strong axioms of infinity and elementary embeddings, to appear.Google Scholar
[8]Scott, D., Measurable cardinals and constructible sets, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 9 (1961) pp. 521–524.Google Scholar