A maximal bounded forcing axiom

Journal of Symbolic Logic 67 (1):130-142 (2002)
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After presenting a general setting in which to look at forcing axioms, we give a hierarchy of generalized bounded forcing axioms that correspond level by level, in consistency strength, with the members of a natural hierarchy of large cardinals below a Mahlo. We give a general construction of models of generalized bounded forcing axioms. Then we consider the bounded forcing axiom for a class of partially ordered sets Γ 1 such that, letting Γ 0 be the class of all stationary-set-preserving partially ordered sets, one can prove the following: (a) $\Gamma_0 \subseteq \Gamma_1$ , (b) Γ 0 = Γ 1 if and only if NS ω 1 is ℵ 1 -dense. (c) If P $\notin \Gamma_1$ , then BFA({P}) fails. We call the bounded forcing axiom for Γ 1 Maximal Bounded Forcing Axiom (MBFA). Finally we prove MBFA consistent relative to the consistency of an inaccessible Σ 2 -correct cardinal which is a limit of strongly compact cardinals



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David Aspero
University of East Anglia

Citations of this work

The Bounded Axiom A Forcing Axiom.Thilo Weinert - 2010 - Mathematical Logic Quarterly 56 (6):659-665.

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Bounded forcing axioms as principles of generic absoluteness.Joan Bagaria - 2000 - Archive for Mathematical Logic 39 (6):393-401.

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