### The Bounded Proper Forcing Axiom

by Goldstern and Shelah. [GoSh:507]

J Symbolic Logic, 1995

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 phi,
``H(chi) models `phi ' '' implies ``exists delta < kappa,
H(delta) models `phi ' ''
We show that BPFA is equivalent to the statement that two
nonisomorphic models of size aleph_1 cannot be made isomorphic
by a proper forcing notion, 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.

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