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Regularity of ultrafilters and the core model. (English) Zbl 0663.03037

This paper contains the following two important results: (1) If there is no model with a measurable cardinal, then every uniform ultrafilter on a singular cardinal is regular. (2) If there is no inner model with a measurable cardinal and \(\kappa >\omega\) is regular with \((\kappa^+)^ K=\kappa^+\), then every uniform ultrafilter on \(\kappa\) is regular. (So in particular, in the core model K, every uniform ultrafilter is regular.)
The author introduces certain combinatorial principles, which are strengthenings of \(\square_{\kappa}\). He shows these hold in K. Then from these combinatorial statements he proves the results about uniform ultrafilters.
Reviewer: N.H.Williams

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
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[1] Benda, M.; Ketonen, J., Regularity of ultrafilters, Isr. J. Math., 17, 231-240 (1974) · Zbl 0312.02055 · doi:10.1007/BF02756872
[2] Chang, C. C.; Asser, G., Extensions of combinatorial principles of Kurepa and Prikry, Theory of Sets and Topology, 49-57 (1972), Berlin: Springer-Verlag, Berlin · Zbl 0269.02037
[3] Chang, C. C.; Keisler, H. J., Model Theory (1973), Amsterdam: North-Holland, Amsterdam · Zbl 0276.02032
[4] A. J. Dodd,The Core Model, London Math. Soc. Lecture Note Series 61, Cambridge University Press, 1982. · Zbl 0474.03027
[5] Dodd, A. J.; Jensen, R. B., The core model, Ann. Math. Logic, 20, 43-75 (1981) · Zbl 0457.03051 · doi:10.1016/0003-4843(81)90011-5
[6] Dodd, A. J.; Jensen, R. B., The covering lemma for K, Ann. Math. Logic, 22, 1-30 (1982) · Zbl 0492.03014 · doi:10.1016/0003-4843(82)90013-4
[7] Donder, H.-D.; Jensen, R. B.; Koppelberg, B. J., Some applications of the core model, Set Theory and Model Theory, Proc. Bonn 1979, 55-97 (1981), Berlin: Springer-Verlag, Berlin · Zbl 0485.03026
[8] R. B. Jensen,Some combinatorial principles of L and V, unpublished manuscript.
[9] Jensen, R. B., The fine structure of the constructible hierarchy, Ann. Math. Logic, 4, 229-308 (1972) · Zbl 0257.02035 · doi:10.1016/0003-4843(72)90001-0
[10] Kanamori, A., Weakly normal filters and irregular ultrafilters, Trans. Am. Math. Soc., 220, 393-399 (1976) · Zbl 0341.02058 · doi:10.2307/1997652
[11] Kanamori, A.; Reinhardt, W.; Solovay, R., Strong axioms of infinity and elementary embeddings, Ann. Math. Logic, 13, 73-116 (1978) · Zbl 0376.02055 · doi:10.1016/0003-4843(78)90031-1
[12] Ketonen, J., Nonregular ultrafilters and large cardinals, Trans. Am. Math. Soc., 224, 61-73 (1976) · Zbl 0352.02045 · doi:10.2307/1997415
[13] Magidor, M., On the existence of nonregular ultrafilters and the cardinality of ultrapowers, Trans. Am. Math. Soc., 249, 97-111 (1979) · Zbl 0409.03032 · doi:10.2307/1998913
[14] Prikry, K., Changing measurable into accessible cardinals, Diss. Math., 68, 5-52 (1970) · Zbl 0212.32404
[15] Prikry, K., On a problem of Gillman and Keisler, Ann. Math. Logic, 2, 179-187 (1970) · Zbl 0209.30701 · doi:10.1016/0003-4843(70)90010-0
[16] Prikry, K., On descendingly complete ultrafilters, 459-488 (1973), Berlin: Springer-Verlag, Berlin · Zbl 0268.02050
[17] P. Welch,Combinatorial Principles in the core model, D. Phil. thesis, Oxford, 1979.
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