The bounded proper forcing axiom

Medientyp: E-Artikel
Titel: The bounded proper forcing axiom
Beteiligte: Goldstern, Martin; Shelah, Saharon
Quelle: Journal of Symbolic Logic ; 60 ( 1995 ) S. 58-73
Erschienen: Cambridge University Press (CUP), 1995
Sprache: Englisch
DOI: 10.2307/2275509
ISSN: 0022-4812; 1943-5886
Schlagwörter: Logic ; Philosophy
Zusammenfassung: <jats:title>Abstract</jats:title><jats:p>The bounded proper forcing axiom BPFA is the statement that for any family of ℵ<jats:sub>1</jats:sub> many maximal antichains of a proper forcing notion, each of size ℵ<jats:sub>1</jats:sub>, there is a directed set meeting all these antichains.</jats:p><jats:p>A regular cardinal <jats:italic>κ</jats:italic> is called ∑<jats:sub>1</jats:sub>-reflecting, if for any regular cardinal <jats:italic>χ</jats:italic>, for all formulas <jats:italic>φ</jats:italic>, “<jats:italic>H</jats:italic>(<jats:italic>χ</jats:italic>) ⊨ ‘<jats:italic>φ</jats:italic>’” implies “∃<jats:italic>δ</jats:italic> < <jats:italic>κ</jats:italic>, <jats:italic>H</jats:italic>(<jats:italic>δ</jats:italic>) ⊨ ‘<jats:italic>φ</jats:italic>’”.</jats:p><jats:p>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 ∑<jats:sub>1</jats:sub>-reflecting cardinal (which is less than the existence of a Mahlo cardinal).</jats:p><jats:p>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.</jats:p>
Beschreibung: <jats:title>Abstract</jats:title><jats:p>The bounded proper forcing axiom BPFA is the statement that for any family of ℵ<jats:sub>1</jats:sub> many maximal antichains of a proper forcing notion, each of size ℵ<jats:sub>1</jats:sub>, there is a directed set meeting all these antichains.</jats:p><jats:p>A regular cardinal <jats:italic>κ</jats:italic> is called ∑<jats:sub>1</jats:sub>-reflecting, if for any regular cardinal <jats:italic>χ</jats:italic>, for all formulas <jats:italic>φ</jats:italic>, “<jats:italic>H</jats:italic>(<jats:italic>χ</jats:italic>) ⊨ ‘<jats:italic>φ</jats:italic>’” implies “∃<jats:italic>δ</jats:italic> < <jats:italic>κ</jats:italic>, <jats:italic>H</jats:italic>(<jats:italic>δ</jats:italic>) ⊨ ‘<jats:italic>φ</jats:italic>’”.</jats:p><jats:p>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 ∑<jats:sub>1</jats:sub>-reflecting cardinal (which is less than the existence of a Mahlo cardinal).</jats:p><jats:p>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.</jats:p>

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