The bounded proper forcing axiom. (English) Zbl 0819.03042

Goldstern, Martin; Shelah, Saharon

J. Symb. Log. 60, No. 1, 58-73 (1995).

Summary: 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 \(\varphi\), “\(H (\chi) \vDash \text{`} \varphi \text{' }\)” implies “\(\exists \delta< \kappa\), \(H(\delta) \vDash \text{`} \varphi \text{' }\)”. We investigate several algebraic consequences of BPFA, 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.

MSC:

03E35	Consistency and independence results
03C55	Set-theoretic model theory
03E55	Large cardinals
03C50	Models with special properties (saturated, rigid, etc.)

Keywords:

existence of isomorphisms between structures; bounded proper forcing axiom; regular cardinal; consistency strength; existence of a \(\Sigma_ 1\)-reflecting cardinal; rigidity of a structure

References:

[1]	DOI: 10.1016/0003-4843(72)90017-4 · Zbl 0255.02069
[2]	DOI: 10.1305/ndjfl/1093634482 · Zbl 0787.03041
[3]	Surveys in set theory 87 pp 1– (1983) · Zbl 0511.00004
[4]	Handbook of set-theoretic topology pp 913– (1984)
[5]	Israel Journal of Mathematics 25 (1976)
[6]	Proper and improper forcing · Zbl 0889.03041
[7]	Proper forcing 940 (1982)
[8]	DOI: 10.1016/0003-4843(78)90009-8 · Zbl 0383.03019
[9]	Axiomatic set theory pp 209– (1984)