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Tall cardinals
Author(s) -
Hamkins Joel D.
Publication year - 2009
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200710084
Subject(s) - regular cardinal , mathematics , forcing (mathematics) , embedding , combinatorics , mathematical analysis , computer science , artificial intelligence
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 ( κ ) > θ and M κ ⊆ 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 2 κ as high as desired. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)