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Hierarchies of forcing axioms I

Published online by Cambridge University Press:  12 March 2014

Itay Neeman  and
Ernest Schimmerling
Itay Neeman
Affiliation:
University of California at los Angeles, Department of Mathematics, Los Angeles, Ca 90095-1555, USA, E-mail: ineeman@math.ucla.edu
Ernest Schimmerling
Affiliation:
Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA 15213-3890, USA, E-mail: eschimme@andrew.cmu.edu
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Abstract

We prove new upper bound theorems on the consistency strengths of SPFA(θ), SPFA(θ-linked) and SPFA(θ+ -cc). Our results are in terms of (θ, Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(ϲ-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper. As a corollary, SPFA(ϲ-linked) and PFA(ϲ-linked) are each equiconsistent with the existence of a -indescribable cardinal. Our upper bound for SPFA(ϲ-c.c) is a -indescribable cardinal, which is consistent with V = L. Our upper bound for SPFA(ϲ+-linked) is a cardinals κ that is (κ+,)-subcompact, which is strictly weaker than κ+-supercompact. The axiom MM(ϲ) is a consequence of SPFA(ϲ+-linked) by a slight refinement of a theorem of Shelah. Our upper bound for SPFA(ϲ++-c.c.) is a cardinal κ that is (κ+, )-subcompact, which is also strictly weaker than κ+-supercompact.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 1 , March 2008 , pp. 343 - 362
Copyright
Copyright © Association for Symbolic Logic 2008

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