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Identity crises and strong compactness III: Woodin cardinals

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Abstract

We show that it is consistent, relative to n ∈ ω supercompact cardinals, for the strongly compact and measurable Woodin cardinals to coincide precisely. In particular, it is consistent for the first n strongly compact cardinals to be the first n measurable Woodin cardinals, with no cardinal above the n th strongly compact cardinal being measurable. In addition, we show that it is consistent, relative to a proper class of supercompact cardinals, for the strongly compact cardinals and the cardinals which are both strong cardinals and Woodin cardinals to coincide precisely. We also show how the techniques employed can be used to prove additional theorems about possible relationships between Woodin cardinals and strongly compact cardinals.

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Authors and Affiliations

  1. Department of Mathematics, Baruch College of CUNY, NewYork, NewYork, 10010, USA

    Arthur W. Apter

  2. Group in Logic and the Methodology of Science, University of California, Berkeley, California, 94720, USA

    Grigor Sargsyan

Authors
  1. Arthur W. Apter
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  2. Grigor Sargsyan
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Correspondence to Arthur W. Apter.

Additional information

The first author's research was partially supported by PSC-CUNY Grant 66489-00-35 and a CUNY Collaborative Incentive Grant.

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Apter, A., Sargsyan, G. Identity crises and strong compactness III: Woodin cardinals. Arch. Math. Logic 45, 307–322 (2006). https://doi.org/10.1007/s00153-005-0316-9

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  • DOI: https://doi.org/10.1007/s00153-005-0316-9

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