Annals of Pure and Applied Logic 89 (2-3):101-115 (1997)

Abstract
We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V “ZFC + Ω is the least inaccessible limit of measurable limits of supercompact cardinals + ƒ : Ω → 2 is a function”, then there is a partial ordering P V so that for , There is a proper class of compact cardinals + If ƒ = 0, then the αth compact cardinal is not supercompact + If ƒ = 1, then the αth compact cardinal is supercompact”. We then prove a generalized version of this theorem assuming κ is a supercompact limit of supercompact cardinals and ƒ : κ → 2 is a function, and we derive as corollaries of the generalized version of the theorem the consistency of the least measurable limit of supercompact cardinals being the same as the least measurable limit of nonsupercompact strongly compact cardinals and the consistency of the least supercompact cardinal being a limit of strongly compact cardinals
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DOI 10.1016/s0168-0072(97)80001-2
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References found in this work BETA

Strong Axioms of Infinity and Elementary Embeddings.Robert M. Solovay - 1978 - Annals of Mathematical Logic 13 (1):73.
On Strong Compactness and Supercompactness.Telis K. Menas - 1975 - Annals of Mathematical Logic 7 (4):327-359.
On the Compactness of ℵ1 and ℵ2.C. A. di Prisco & J. Henle - 1978 - Journal of Symbolic Logic 43 (3):394-401.

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Citations of this work BETA

The Lottery Preparation.Joel David Hamkins - 2000 - Annals of Pure and Applied Logic 101 (2-3):103-146.
Identity Crises and Strong Compactness.Arthur W. Apter & James Cummings - 2000 - Journal of Symbolic Logic 65 (4):1895-1910.
Failures of SCH and Level by Level Equivalence.Arthur W. Apter - 2006 - Archive for Mathematical Logic 45 (7):831-838.

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