# Sh:507

• Goldstern, M., & Shelah, S. (1995). The bounded proper forcing axiom. J. Symbolic Logic, 60(1), 58–73.
• Abstract:
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)\models \varphi’ ” implies “\exists\delta<\kappa, H(\delta)\models \varphi’ ”

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.

• Version 1996-03-19_10 (32p) published version (17p)
Bib entry
@article{Sh:507,
author = {Goldstern, Martin and Shelah, Saharon},
title = {{The bounded proper forcing axiom}},
journal = {J. Symbolic Logic},
fjournal = {The Journal of Symbolic Logic},
volume = {60},
number = {1},
year = {1995},
pages = {58--73},
issn = {0022-4812},
mrnumber = {1324501},
mrclass = {03E35 (03E55)},
doi = {10.2307/2275509},
note = {\href{https://arxiv.org/abs/math/9501222}{arXiv: math/9501222}},
arxiv_number = {math/9501222}
}