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Laver sequences for extendible and super-almost-huge cardinals

Published online by Cambridge University Press:  12 March 2014

Paul Corazza
Paul Corazza*
Affiliation:
310 E. Washington Avenue, Fairfield, IA 52556, USA E-mail: [email protected]
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Abstract

Versions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses. Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point κ, and there is an inaccessible above κ, then it is consistent for there to be a regular class that admits no Laver sequence. We also show that extendible cardinals are Laver-generating, i.e., that assuming only that κ is extendible, there is an extendible Laver sequence at κ. We use the method of proof to answer a question about Laver-closure of extendible cardinals at inaccessibles. Finally, we consider Laver sequences for super-almost-huge cardinals. Assuming slightly more than super-almost-hugeness, we show that there are super-almost-huge Laver sequences, improving the previously known upper bound for such Laver sequences. We also describe conditions under which the canonical construction of a Laver sequence fails for super-almost-huge cardinals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 3 , September 1999 , pp. 963 - 983
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1] Barbanel, J., Flipping properties and huge cardinals, Fundamenta Mathematicae, vol. 13 (1989), pp. 171–188.CrossRefGoogle Scholar
[2] Corazza, P., The wholeness axiom and Laver sequences, to appear in Annals of Pure and Applied Logic.Google Scholar
[3] Corazza, P., A new large cardinal and Laver sequences for extendibles, Fundamenta Mathematicae, vol. 152 (1997), pp. 183–188.Google Scholar
[4] Devlin, K., The Yorkshireman's guide to proper forcing, Surveys in set theory (Mathias, A., editor), Cambridge University Press, 1983, pp. 60–115.CrossRefGoogle Scholar
[5] Enayat, A., Undefinable classes and definable elements in models of set theory and arithmetic, Proceedings of the American Mathematical Society, vol. 103 (1988), pp. 1216–1220.CrossRefGoogle Scholar
[6] Gitik, M. and Shelah, S., On certain indestructibility of strong cardinals and a question of Hajnal, Archive of Mathematical Logic, vol. 28 (1989), pp. 35–42.CrossRefGoogle Scholar
[7] Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[8] Kanamori, A., The higher infinite, Springer-Verlag, New York, 1994.Google Scholar
[9] Kunen, K., Elementary embeddings and infinitary combinatorics, this Journal, vol. 36 (1971), pp. 407–413.Google Scholar
[10] Kunen, K., Set theory: an introduction to independence proofs, North Holland, New York, 1980.Google Scholar
[11] Laver, R., Making the supercompactness ofn indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), no. 4, pp. 385–388.CrossRefGoogle Scholar
[12] Laver, R., The left distributive law andfreeness of an algebra of elementary embeddings, Advances in Mathematics, vol. 91 (1992), pp. 209–231.CrossRefGoogle Scholar
[13] Martin, D. and Steel, J., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125.CrossRefGoogle Scholar
[14] Menas, T., Consistency results concerning supercompactness, Transactions of the American Mathematical Society, vol. 223 (1976), pp. 61–91.CrossRefGoogle Scholar
[15] Solovay, R., Reinhardt, W., and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73–116.CrossRefGoogle Scholar