The bounded proper forcing axiom

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Abstract

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 ∑ 1 -reflecting, if for any regular cardinal χ , for all formulas φ , “ H ( χ ) ⊨ ‘ φ ’” implies “∃ δ < κ , H ( δ ) ⊨ ‘ φ ’”. 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 ∑ 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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