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On the least strongly compact cardinal

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Abstract

We prove that under the assumption of a supercompact cardinal κ which is a limit of supercompact cardinals, for any increasing Σ2 function φ the set {∂<κ:∂ is at least φ(∂) supercompact, is strongly compact, yet is not fully supercompact} is unbounded in κ. We then use ideas of Magidor to show that under the hypotheses of a supercompact cardinal which is a limit of supercompact cardinals it is consistent for the least strongly compact cardinal κ0 to be at least φ(κ0) supercompact yet not to be fully supercompact, where φ is again an increasing Σ2 function which also meets certain other technical restrictions.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, Mass., USA

    Arthur W. Apter

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  1. Arthur W. Apter
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Additional information

The author wishes to thank Menachem Magidor for helpful conversations and suggestions in method which were used in the proof of Theorem 2.

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Apter, A.W. On the least strongly compact cardinal. Israel J. Math. 35, 225–233 (1980). https://doi.org/10.1007/BF02761194

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  • DOI: https://doi.org/10.1007/BF02761194

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