Project Euclid

The bounded proper forcing axiom BPFA is the statement that for any family of $\aleph_1$ many maximal antichains of a proper forcing notion, each of size $\aleph_1$, there is a directed set meeting all these antichains. A regular cardinal $\kappa$ is called $\Sigma_1$-reflecting, if for any regular cardinal $\chi$, for all formulas $\varphi, "H(\chi) \models`\varphi'"$ implies "$\exists\delta < \kappa, H(\delta) \models`\varphi'"$. 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 $\Sigma_1$-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.