Goldstern, Martin; Shelah, Saharon The bounded proper forcing axiom. (English) Zbl 0819.03042 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. Cited in 3 ReviewsCited in 25 Documents 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 PDF BibTeX XML Cite \textit{M. Goldstern} and \textit{S. Shelah}, J. Symb. Log. 60, No. 1, 58--73 (1995; Zbl 0819.03042) Full Text: DOI arXiv OpenURL 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) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.