The Consistency Strength of $$\aleph{\omega}$$ and $$\aleph{{\omega}_1}$$ Being Rowbottom Cardinals Without the Axiom of Choice

Archive for Mathematical Logic 45 (6):721-737 (2006)
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Abstract

We show that for all natural numbers n, the theory “ZF + DC $_{\aleph_n}$ + $\aleph_{\omega}$ is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + $\aleph_{\omega_1}$ is an ω 2-Rowbottom cardinal carrying an ω 2-Rowbottom filter and ω 1 is regular” has the same consistency strength as the theory “ZFC + There exist ω 1 measurable cardinals”. We also discuss some generalizations of these results

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Citations of this work

Making all cardinals almost Ramsey.Arthur W. Apter & Peter Koepke - 2008 - Archive for Mathematical Logic 47 (7-8):769-783.
The consistency strength of choiceless failures of SCH.Arthur W. Apter & Peter Koepke - 2010 - Journal of Symbolic Logic 75 (3):1066-1080.

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References found in this work

The core model.A. Dodd & R. Jensen - 1981 - Annals of Mathematical Logic 20 (1):43-75.
The covering lemma for L[U].A. J. Dodd & R. B. Jensen - 1982 - Annals of Mathematical Logic 22 (2):127-135.
The consistency strength of the free-subset property for ωω.Peter Koepke - 1984 - Journal of Symbolic Logic 49 (4):1198 - 1204.
Successive weakly compact or singular cardinals.Ralf-Dieter Schindler - 1999 - Journal of Symbolic Logic 64 (1):139-146.
Some applications of short core models.Peter Koepke - 1988 - Annals of Pure and Applied Logic 37 (2):179-204.

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