On the existence of nonregular ultrafilters and the cardinality of ultrapowers

Magidor, Menachem

doi:10.1090/S0002-9947-1979-0526312-2

On the existence of nonregular ultrafilters and the cardinality of ultrapowers
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by Menachem Magidor PDF

Trans. Amer. Math. Soc. 249 (1979), 97-111 Request permission

Abstract:

Assuming the consistency of huge cardinals, we prove that ${\omega _3}$ can carry an ultrafilter D such that ${\omega _1}^{{\omega _3}}/D$ has cardinality ${\omega _3}$. (Hence D is not $({\omega _3}, {\omega _1})$ regular.) Similarly ${\omega _2}$ can carry an ultrafilter D such that ${\omega ^{{\omega _2}}}/D$ has cardinality ${\omega _2}$. (Hence D is not $({\omega _2}, \omega )$ regular.)

References

J. L. Bell and A. B. Slomson, Models and ultraproducts: An introduction, North-Holland Publishing Co., Amsterdam-London, 1969. MR 0269486
Miroslav Benda and Jussi Ketonen, Regularity of ultrafilters, Israel J. Math. 17 (1974), 231–240. MR 396264, DOI 10.1007/BF02756872
P. Erdős and A. Hajnal, Unsolved problems in set theory, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 17–48. MR 0280381
T. Frayne, A. C. Morel, and D. S. Scott, Reduced direct products, Fund. Math. 51 (1962/63), 195–228. MR 142459, DOI 10.4064/fm-51-3-195-228
Thomas J. Jech, Lectures in set theory, with particular emphasis on the method of forcing, Lecture Notes in Mathematics, Vol. 217, Springer-Verlag, Berlin-New York, 1971. MR 0321738
A. Kanamori, Weakly normal filters and irregular ultrafilters, Trans. Amer. Math. Soc. 220 (1976), 393–399. MR 480041, DOI 10.1090/S0002-9947-1976-0480041-X
Robert M. Solovay, William N. Reinhardt, and Akihiro Kanamori, Strong axioms of infinity and elementary embeddings, Ann. Math. Logic 13 (1978), no. 1, 73–116. MR 482431, DOI 10.1016/0003-4843(78)90031-1
H. Jerome Keisler, On cardinalities of ultraproducts, Bull. Amer. Math. Soc. 70 (1964), 644–647. MR 166106, DOI 10.1090/S0002-9904-1964-11219-2

A survey of ultraproducts in logic

Jussi Ketonen, Nonregular ultrafilters and large cardinals, Trans. Amer. Math. Soc. 224 (1976), 61–73. MR 419236, DOI 10.1090/S0002-9947-1976-0419236-X
Kenneth Kunen, Saturated ideals, J. Symbolic Logic 43 (1978), no. 1, 65–76. MR 495118, DOI 10.2307/2271949
K. Kunen and J. B. Paris, Boolean extensions and measurable cardinals, Ann. Math. Logic 2 (1970/71), no. 4, 359–377. MR 277381, DOI 10.1016/0003-4843(71)90001-5
Menachem Magidor, On the singular cardinals problem. I, Israel J. Math. 28 (1977), no. 1-2, 1–31. MR 491183, DOI 10.1007/BF02759779
Karel Prikry, On a problem of Gillman and Keisler, Ann. Math. Logic 2 (1970), no. 2, 179–187. MR 269513, DOI 10.1016/0003-4843(70)90010-0

G.C.H. and large cardinals

Robert M. Solovay, Real-valued measurable cardinals, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 397–428. MR 0290961
R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin’s problem, Ann. of Math. (2) 94 (1971), 201–245. MR 294139, DOI 10.2307/1970860