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 the consistency strength of the bounded proper forcing axiom is exactly the existence of a §1-re∞ecting cardinal (which is less than the ex- istence 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 rigid- ity of a structure.
[1]
Saharon Shelah,et al.
Models with second order properties II. Trees with no undefined branches
,
1978
.
[2]
S. Shelah,et al.
Annals of Pure and Applied Logic
,
1991
.
[3]
William Mitchell,et al.
Aronszajn trees and the independence of the transfer property
,
1972
.
[4]
Saharon Shelah,et al.
Refuting ehrenfeucht conjecture on rigid models
,
1976
.
[5]
James E. Baumgartner,et al.
Applications of the Proper Forcing Axiom
,
1984
.
[6]
Sakaé Fuchino.
On Potential Embedding and Versions of Martin's Axiom
,
1992,
Notre Dame J. Formal Log..