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Supercompact cardinals, elementary embeddings and fixed points1

Published online by Cambridge University Press:  12 March 2014

Julius B. Barbanel
Julius B. Barbanel*
Affiliation:
Union College, Schnectady New York 12308
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Extract

Supercompactness is usually defined in terms of the existence of certain ultrafilters. By the well-known procedure of taking ultrapowers of V (the universe of sets) and transitive collapses, one obtains transitive inner models of V and corresponding elementary embeddings from V into these inner models. These embeddings have been studied extensively (see, e.g. [3] or [4]). We investigate the action of these embeddings on cardinals. In particular, we establish a characterization, based upon cofinality, of which cardinals are fixed by these embeddings.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 1 , March 1982 , pp. 84 - 88
Copyright
Copyright © Association for Symbolic Logic 1982

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Footnotes

1

This paper is part of the author's Ph.D. thesis [1] written at the State University of New York at Buffalo under the supervision of Professor Nicolas Goodman to whom the author is grateful.

References

REFERENCES

[1]Barbanel, J. B., Results on supercompact cardinals, Ph.D. Thesis, State University of New York at Buffalo, 1979.Google Scholar
[2]Magidor, M., Private communication.Google Scholar
[3]Solovay, R. M.Strongly compact cardinals and the G.C. H., Proceedings of the Tarski Symosium (Henkin, L.et al., Editors), Proceedings of Symposia in Pure Mathematics, vol. 25, American Mathematical Society, Providence, R.I., 1974, pp. 365–372.CrossRefGoogle Scholar
[4]Solovay, R. M., Reinhardt, W. N. and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13, No. 1 (1978), pp. 73–116.CrossRefGoogle Scholar