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NORMAL MEASURES ON A TALL CARDINAL

Published online by Cambridge University Press:  13 February 2019

ARTHUR W. APTER  and
JAMES CUMMINGS
ARTHUR W. APTER
Affiliation:
DEPARTMENT OF MATHEMATICS BARUCH COLLEGE, CITY UNIVERSITY OF NEW YORK NEW YORK, NY 10010, USA and DEPARTMENT OF MATHEMATICS CUNY GRADUATE CENTER, 365 FIFTH AVENUE NEW YORK, NY 10016, USAE-mail: [email protected]: http://faculty.baruch.cuny.edu/aapter
JAMES CUMMINGS
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA 15213, USAE-mail: [email protected]: http://www.math.cmu.edu/math/faculty/cummings.html
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Abstract

We study the number of normal measures on a tall cardinal. Our main results are that:

  • • The least tall cardinal may coincide with the least measurable cardinal and carry as many normal measures as desired.

  • • The least measurable limit of tall cardinals may carry as many normal measures as desired.

Keywords

Primary 03E35Secondary 03E55tall cardinalnormal measurenonstationary support iteration
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 1 , March 2019 , pp. 178 - 204
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

REFERENCES

Apter, A. W., Some remarks on indestructibility and Hamkins’ lottery preparation. Archive for Mathematical Logic, vol. 42 (2003), pp. 717–735.10.1007/s00153-003-0181-3CrossRefGoogle Scholar
Apter, A. W. and Cummings, J., Identity crises and strong compactness, this Journal, vol. 65 (2000), pp. 1895–1910.Google Scholar
Apter, A. W. and Gitik, M., On tall cardinals and some related generalizations. Israel Journal of Mathematics, vol. 202 (2014), pp. 343–373.10.1007/s11856-014-1073-4CrossRefGoogle Scholar
Beller, A., Jensen, R., and Welch, P., Coding the Universe, London Mathematical Society Lecture Note Series, vol. 47, Cambridge University Press, Cambridge, 1982.Google Scholar
Friedman, S.-D. and Magidor, M., The number of normal measures, this Journal, vol. 74 (2009), pp. 1069–1080.Google Scholar
Friedman, S.-D. and Thompson, K., Perfect trees and elementary embeddings, this Journal, vol. 74 (2008), pp. 906–918.Google Scholar
Gitik, M. and Shelah, S., On certain indestructibility of strong cardinals and a question of Hajnal. Archive for Mathematical Logic, vol. 28 (1989), pp. 35–42.10.1007/BF01624081CrossRefGoogle Scholar
Hamkins, J. D., The lottery preparation. Annals of Pure and Applied Logic, vol. 101 (2000),pp. 103–146.10.1016/S0168-0072(99)00010-XCrossRefGoogle Scholar
Hamkins, J. D., Gap forcing. Israel Journal of Mathematics, vol. 125 (2001), pp. 237–252.10.1007/BF02773382CrossRefGoogle Scholar
Hamkins, J. D., Tall cardinals. Mathematical Logic Quarterly, vol. 55 (2009), pp. 68–86.10.1002/malq.200710084CrossRefGoogle Scholar
Jensen, R. B. and Steel, J. R., K without the measurable, this Journal, vol. 78 (2013), pp. 708–734.Google Scholar
Kunen, K., Some applications of iterated ultrapowers in set theory. Annals of Mathematical Logic, vol. 1 (1970), pp. 179–227.10.1016/0003-4843(70)90013-6CrossRefGoogle Scholar
Magidor, M., How large is the first strongly compact cardinal? or a study on identity crises. Annals of Mathematical Logic, vol. 10 (1976), pp. 33–57.10.1016/0003-4843(76)90024-3CrossRefGoogle Scholar
Schindler, R.-D., personal communication.Google Scholar
Schindler, R.-D., Iterates of the core model, this Journal, vol. 71 (2006), pp. 241–251.Google Scholar
Solovay, R. M., Reinhardt, W. N., and Kanamori, A., Strong axioms of infinity and elementary embeddings. Annals of Mathematical Logic, vol. 13 (1978), pp. 73–116.10.1016/0003-4843(78)90031-1CrossRefGoogle Scholar
Steel, J. R., An outline of inner model theory, Handbook of Set Theory (Kanamori, A. and Foreman, M., editors), Springer, Dordrecht, 2010, pp. 1595–1684.10.1007/978-1-4020-5764-9_20CrossRefGoogle Scholar
Villaveces, A., Chains of end elementary extensions of models of set theory, this Journal, vol. 63 (1998), pp. 1116–1136.Google Scholar