# Sh:507

- Goldstern, M., & Shelah, S. (1995).
*The bounded proper forcing axiom*. J. Symbolic Logic,**60**(1), 58–73. arXiv: math/9501222 DOI: 10.2307/2275509 MR: 1324501 -
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.

- 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}, doi = {10.2307/2275509}, mrclass = {03E35 (03E55)}, mrnumber = {1324501}, mrreviewer = {James Baumgartner}, doi = {10.2307/2275509}, note = {\href{https://arxiv.org/abs/math/9501222}{arXiv: math/9501222}}, arxiv_number = {math/9501222} }