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Forcing the failure of ch by adding a real

Saharon Shelah & Hugh Woodin

Journal of Symbolic Logic 49 (4):1185-1189 (1984)

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Combinatorial properties of classical forcing notions.Jörg Brendle - 1995 - Annals of Pure and Applied Logic 73 (2):143-170.details We investigate the effect of adding a single real on cardinal invariants associated with the continuum. We show:1. adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size ω1;2. Laver and Mathias forcing collapse the dominating number to ω1, and thus two Laver or Mathias reals added iteratively always force CH;3. Miller's rational perfect set forcing preserves the axiom MA. Download Export citation Bookmark 3 citations
Shelah's strong covering property and CH in V [r ].Esfandiar Eslami & Mohammad Golshani - 2012 - Mathematical Logic Quarterly 58 (3):153-158.details In this paper we review Shelah's strong covering property and its applications. We also extend some of the results of Shelah and Woodin on the failure of equation image by adding a real. Download Export citation Bookmark 1 citation
Complexity of reals in inner models of set theory.Boban Velickovic & Hugh Woodin - 1998 - Annals of Pure and Applied Logic 92 (3):283-295.details We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals are an uncountable Fσ set (...) Download Export citation Bookmark 2 citations
(1 other version)The negation of the singular cardinal hypothesis from o(K)=K++.Moti Gitik - 1989 - Annals of Pure and Applied Logic 43 (3):209-234.details Download Export citation Bookmark 39 citations
(1 other version)The negation of the singular cardinal hypothesis from< i> o(< i> K_)=< i> K< sup>++.Moti Gitik - 1989 - Annals of Pure and Applied Logic 43 (3):209-234.details Download Export citation Bookmark 30 citations
Complexity of reals in inner models of set theory.Boban Velickovic & W. Hugh Woodin - 1998 - Annals of Pure and Applied Logic 92 (3):283-295.details We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals are an uncountable Fσ set (...) Download Export citation Bookmark 3 citations
On the Mitchell and Rudin-Kiesler orderings of ultrafilters.Moti Gitik - 1988 - Annals of Pure and Applied Logic 39 (2):175-197.details Download Export citation Bookmark 5 citations
Minimal collapsing extensions of models of zfc.Lev Bukovský & Eva Copláková-Hartová - 1990 - Annals of Pure and Applied Logic 46 (3):265-298.details Download Export citation Bookmark 8 citations

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