Tall cardinals

Mathematical Logic Quarterly 55 (1):68-86 (2009)

Authors
Joel David Hamkins
Oxford University
Abstract
A cardinal κ is tall if for every ordinal θ there is an embedding j: V → M with critical point κ such that j > θ and Mκ ⊆ M. Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a strong cardinal. Any tall cardinal κ can be made indestructible by a variety of forcing notions, including forcing that pumps up the value of 2κ as high as desired
Keywords strong cardinals  Large cardinals  forcing  GCH  tall cardinals
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DOI 10.1002/malq.200710084
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References found in this work BETA

The Lottery Preparation.Joel David Hamkins - 2000 - Annals of Pure and Applied Logic 101 (2-3):103-146.
The Negation of the Singular Cardinal Hypothesis From o=K++.Moti Gitik - 1989 - Annals of Pure and Applied Logic 43 (3):209-234.
Identity Crises and Strong Compactness.Arthur W. Apter & James Cummings - 2000 - Journal of Symbolic Logic 65 (4):1895-1910.

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Citations of this work BETA

The Large Cardinals Between Supercompact and Almost-Huge.Norman Lewis Perlmutter - 2015 - Archive for Mathematical Logic 54 (3-4):257-289.
Tallness and Level by Level Equivalence and Inequivalence.Arthur W. Apter - 2010 - Mathematical Logic Quarterly 56 (1):4-12.
Normal Measures on a Tall Cardinal.Arthur W. Apter & James Cummings - 2019 - Journal of Symbolic Logic 84 (1):178-204.
Large Cardinals Need Not Be Large in HOD.Yong Cheng, Sy-David Friedman & Joel David Hamkins - 2015 - Annals of Pure and Applied Logic 166 (11):1186-1198.
On Extensions of Supercompactness.Robert S. Lubarsky & Norman Lewis Perlmutter - 2015 - Mathematical Logic Quarterly 61 (3):217-223.

View all 11 citations / Add more citations

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