On the least strongly compact cardinal |
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Authors: | Arthur W Apter |
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Institution: | (1) Department of Mathematics, Massachusetts Institute of Technology, 02139 Cambridge, Mass., USA;(2) Department of Mathematics, University of Miami, 33124 Coral Gables, FL, USA |
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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.
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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