On the existence of nonregular ultrafilters and the cardinality of ultrapowers
Author:
Menachem Magidor
Journal:
Trans. Amer. Math. Soc. 249 (1979), 97111
MSC:
Primary 03E05; Secondary 03E35, 03E55
MathSciNet review:
526312
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Additional Information
Abstract: Assuming the consistency of huge cardinals, we prove that can carry an ultrafilter D such that has cardinality . (Hence D is not regular.) Similarly can carry an ultrafilter D such that has cardinality . (Hence D is not regular.)
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 J. L. Bell and A. B. Slomson, Models and ultraproducts. An introduction, North Holland, Amsterdam, 1969. MR 0269486 (42:4381)
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 P. Erdös and A. Hajnal, Unsolved problems in set theory, Axiomatic Set Theory, Proc. Sympos. Pure Math., vol. 13, part I, Amer. Math. Soc., Providence, R. I., 1971, pp. 1748. MR 0280381 (43:6101)
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 A. Kanamori, Weakly normal filters and irregular ultrafilters, Trans. Amer. Math. Soc. 220 (1976), 393399. MR 0480041 (58:240)
 [7]
 A. Kanamori, W. N. Reinhardt and R. M. Solovay, Elementary embeddings and strong axioms of infinity, Ann. Math. Logic 13 (1978), 73116. MR 482431 (80h:03072)
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 H. J. Keisler, On cardinalities of ultraproducts, Bull. Amer. Math. Soc. 70 (1964), 644647. MR 0166106 (29:3384)
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 , A survey of ultraproducts in logic, Methodology and Philosophy of Science, Proc. of the 1964 Internat. Congress, Amsterdam, 1972, pp. 112125.
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 J. Ketonen, Nonregular ultrafilters and large cardinals, Trans. Amer. Math. Soc. 224 (1976), 6173. MR 0419236 (54:7260)
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 K. Kunen, On saturated ideals, J. Symbolic Logic 43 (1978), 6576. MR 495118 (80a:03068)
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 K. Kunen and J. H. Paris, Boolean extensions and measurable cardinals, Ann. Math. Logic 2 (1971), 359377. MR 0277381 (43:3114)
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 M. Magidor, On the singular cardinals problem. II, Ann. of Math. 106 (1977), 514547. MR 0491184 (58:10449b)
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 K. Prikry, On a problem of Gillman and Keisler, Ann. Math. Logic 2 (1970), 179187. MR 0269513 (42:4408)
 [15]
 J. Silver, G.C.H. and large cardinals (to appear).
 [16]
 R. M. Solovay, Real valued measurable cardinals, Axiomatic Set Theory, Proc. Sympos. Pure Math., vol. 13, part I, Amer. Math. Soc., Providence, R. I., 1971, pp. 397428. MR 0290961 (45:55)
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 R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin problem, Ann. of Math. (2) 94 (1971), 201245. MR 0294139 (45:3212)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197905263122
PII:
S 00029947(1979)05263122
Keywords:
Ultrafilter,
ultraproduct,
regular ultrafilter,
huge cardinals
Article copyright:
© Copyright 1979
American Mathematical Society
