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Square in Core Models

Published online by Cambridge University Press:  15 January 2014

Ernest Schimmerling  and
Martin Zeman
Ernest Schimmerling
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA. E-mail: [email protected]
Martin Zeman
Affiliation:
Institut Für Formale Logik, Währinger Straße 25, A-1090 Wien, Austria. Mathematical Institute Sav, Štefánikova 49, 814 73 Bratislava, Slovakia. E-mail: [email protected]
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Abstract

We prove that in all Mitchell-Steel core models, □k holds for all k. (See Theorem 2.) From this we obtain new consistency strength lower bounds for the failure of □k if k is either singular and countably closed, weakly compact, or measurable. (Corollaries 5, 8, and 9.) Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, □k holds iff k is not subcompact. (See Theorem 15; the only if direction is essentially due to Jensen.)

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 7 , Issue 3 , September 2001 , pp. 305 - 314
Copyright
Copyright © Association for Symbolic Logic 2001

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