# The Bounded Proper Forcing Axiom

@article{Goldstern1995TheBP, title={The Bounded Proper Forcing Axiom}, author={Martin Goldstern and Saharon Shelah}, journal={J. Symb. Log.}, year={1995}, volume={60}, pages={58-73} }

The bounded proper forcing axiom BPFA is the statement that for any family of @1 many maximal antichains of a proper forcing notion, each of size @1, there is a directed set meeting all these antichains. A regular cardinalis called §1-re∞ecting, if for any regular cardinal ´, for all formulas ', \H(´) j= '''" implies \9-<•, H(-)j= '''" We show that BPFA is equivalent to the statement that two nonisomorphic models of size @1 cannot be made isomorphic by a proper forcing notion, and we show that…

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## 71 Citations

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- Computer Science, MathematicsJournal of Symbolic Logic
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A hierarchy of generalized bounded forcing axioms that correspond level by level, in consistency strength, with the members of a natural hierarchy of large cardinals below a Mahlo are given.

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We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(ω_1) which is Δ_1 definable with parameter a subset of ω_1. Our proof shows that if BPFA holds then any…

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