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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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