The bounded proper forcing axiom

Journal of Symbolic Logic 60 (1):58-73 (1995)
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Abstract

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ 1 many maximal antichains of a proper forcing notion, each of size ℵ 1 , there is a directed set meeting all these antichains. A regular cardinal κ is called Σ 1 -reflecting, if for any regular cardinal χ, for all formulas $\varphi, "H(\chi) \models`\varphi'"$ implies " $\exists\delta . We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the bounded proper forcing axiom is exactly the existence of a Σ 1 -reflecting cardinal (which is less than the existence of a Mahlo cardinal). We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure

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References found in this work

On potential embedding and versions of Martin's axiom.Sakaé Fuchino - 1992 - Notre Dame Journal of Formal Logic 33 (4):481-492.
Proper and Improper Forcing.Péter Komjáath - 2000 - Bulletin of Symbolic Logic 6 (1):83-86.

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