March 2008 Hierarchies of forcing axioms I
Itay Neeman, Ernest Schimmerling
J. Symbolic Logic 73(1): 343-362 (March 2008). DOI: 10.2178/jsl/1208358756


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(𝔠-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper. As a corollary, SPFA(𝔠-linked) and PFA(𝔠-linked) are each equiconsistent with the existence of a Σ21-indescribable cardinal. Our upper bound for SPFA(𝔠-c.c.) is a Σ22-indescribable cardinal, which is consistent with V=L. Our upper bound for SPFA(𝔠+-linked) is a cardinal κ that is (κ+, Σ21)-subcompact, which is strictly weaker than κ+-supercompact. The axiom MM(𝔠) is a consequence of SPFA(𝔠+-linked) by a slight refinement of a theorem of Shelah. Our upper bound for SPFA(𝔠++-c.c.) is a cardinal κ that is (κ+, Σ22)-subcompact, which is also strictly weaker than κ+-supercompact.


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Itay Neeman. Ernest Schimmerling. "Hierarchies of forcing axioms I." J. Symbolic Logic 73 (1) 343 - 362, March 2008.


Published: March 2008
First available in Project Euclid: 16 April 2008

zbMATH: 1154.03032
MathSciNet: MR2387946
Digital Object Identifier: 10.2178/jsl/1208358756

Rights: Copyright © 2008 Association for Symbolic Logic


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Vol.73 • No. 1 • March 2008
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