## The bounded proper forcing axiom.(English)Zbl 0819.03042

Summary: 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 $$\varphi$$, “$$H (\chi) \vDash \text{} \varphi \text{' }$$” implies “$$\exists \delta< \kappa$$, $$H(\delta) \vDash \text{} \varphi \text{' }$$”. 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.

### MSC:

 03E35 Consistency and independence results 03C55 Set-theoretic model theory 03E55 Large cardinals 03C50 Models with special properties (saturated, rigid, etc.)
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### References:

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