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A maximal bounded forcing axiom

Published online by Cambridge University Press:  12 March 2014

David Asperó
David Asperó*
Affiliation:
Departament De Lògica, Història I Filosofia De La Ciència, Universitat De Barcelona, Baldiri I Reixach, S/N, 08028 Barcelona, Catalonia, Spain, E-mail: [email protected]
*
Institut für formale Logik, Universität Wien, Währingerstrasse 25, 1090 Wien, Austria, [email protected]
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Abstract

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 Γ1 such that, letting Γ0 be the class of all stationary-set-preserving partially ordered sets, one can prove the following:

(a) Γ0 ⊆ Γ1,

(b) Γ0 = Γ1 if and only if NSω1 is ℵ1-dense.

(c) If P ∉ Γ1, then BFA({P}) fails.

We call the bounded forcing axiom for Γ1Maximal Bounded Forcing Axiom (MBFA). Finally we prove MBFA consistent relative to the consistency of an inaccessible Σ2-correct cardinal which is a limit of strongly compact cardinals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 130 - 142
Copyright
Copyright © Association for Symbolic Logic 2002

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References

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