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

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

DOI: https://doi.org/10.1090/S0002-9947-1979-0526312-2

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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.)

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