THE BOUNDED PROPER FORCING AXIOM

The bounded proper forcing axiom BPFA is the statement that for any family of N1 many maximal antichains of a proper forcing notion, each of size N1, there is a directed set meeting all these antichains. A regular cardinal kappa is called SIGMA1-reflecting, if for any regular cardinal CHI, for all formulas phi, ''H(CHI) ??'phi''' implies ''??delta < kappa, H(delta) ??'phi'''. 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 SIGMA1-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.