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The Consistency Strength of \(\aleph_{\omega}\) and \(\aleph_{{\omega}_1}\) Being Rowbottom Cardinals Without the Axiom of Choice

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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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Authors and Affiliations

  1. Department of Mathematics, Baruch College of CUNY, New York, NY, 10010, USA

    Arthur W. Apter

  2. The CUNY Graduate Center, Mathematics, 365 Fifth Avenue, New York, NY, 10016, USA

    Arthur W. Apter

  3. Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität, 53115, Bonn, Germany

    Peter Koepke

Authors
  1. Arthur W. Apter
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  2. Peter Koepke
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Correspondence to Peter Koepke.

Additional information

The first author’s research was partially supported by PSC-CUNY Grant 66489-00-35 and a CUNY Collaborative Incentive Grant. In addition, the first author wishes to thank the members of the set theory group in Bonn for all of the hospitality shown him during his visits to the Mathematisches Institut.

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Apter, A.W., Koepke, P. The Consistency Strength of \(\aleph_{\omega}\) and \(\aleph_{{\omega}_1}\) Being Rowbottom Cardinals Without the Axiom of Choice. Arch. Math. Logic 45, 721–737 (2006). https://doi.org/10.1007/s00153-006-0005-3

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  • DOI: https://doi.org/10.1007/s00153-006-0005-3

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