Journal of Symbolic Logic

Hierarchies of forcing axioms I

Itay Neeman and Ernest Schimmerling

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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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J. Symbolic Logic Volume 73, Issue 1 (2008), 343-362.

First available in Project Euclid: 16 April 2008

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Neeman, Itay; Schimmerling, Ernest. Hierarchies of forcing axioms I. J. Symbolic Logic 73 (2008), no. 1, 343--362. doi:10.2178/jsl/1208358756.

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