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The bounded proper forcing axiom

Published online by Cambridge University Press:  12 March 2014

Martin Goldstern
Affiliation:
Institut für Algebra und Diskrete Mathematik, Technische Universität Wien, A-1040 Wien, Austria, E-mail: goldstrn@rsmb.tuwien.ac.at
Saharon Shelah
Affiliation:
Department of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel, E-mail: shelah@math.huji.ac.il

Abstract

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ1 many maximal antichains of a proper forcing notion, each of size ℵ1, 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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1995

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References

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