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PRESERVING NONNULL WITH SUSLIN + FORCINGS
, 2005
"... Abstract. We introduce the notion of effective Axiom A and use it to show that some popular tree forcings are Suslin +. We introduce transitive nep and present a simplified version of Shelah’s “preserving a little implies preserving much”: If I is a Suslin ccc ideal (e.g. Lebesguenull or meager) an ..."
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Abstract. We introduce the notion of effective Axiom A and use it to show that some popular tree forcings are Suslin +. We introduce transitive nep and present a simplified version of Shelah’s “preserving a little implies preserving much”: If I is a Suslin ccc ideal (e.g. Lebesguenull or meager) and P is a transitive nep forcing (e.g. P is Suslin +) and P doesn’t make any Ipositive Borel set small, then P doesn’t make any Ipositive set small. 1.
Generic absoluteness under projective forcing
"... We study the preservation of the property of L(R) being a Solovay model under projective ccc forcing extensions. We compute the exact consistency strength of the generic absoluteness of L(R) under forcing with projective ccc partial orderings and, as an application, we build models in which Martin’s ..."
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We study the preservation of the property of L(R) being a Solovay model under projective ccc forcing extensions. We compute the exact consistency strength of the generic absoluteness of L(R) under forcing with projective ccc partial orderings and, as an application, we build models in which Martin’s Axiom holds for Σ 1 n partial orderings, but it fails for the
Archive for Mathematical Logic manuscript No. (will be inserted by the editor)
"... Abstract. We study the preservation of the property of L(R) being a Solovay model under ..."
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Abstract. We study the preservation of the property of L(R) being a Solovay model under
THIN EQUIVALENCE RELATIONS AND INNER MODELS
"... Abstract. We describe the inner models with representatives in all equivalence classes of thin equivalence relations in a given projective pointclass of even level assuming projective determinacy. These models are characterized by their correctness and the property that they correctly compute the tr ..."
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Abstract. We describe the inner models with representatives in all equivalence classes of thin equivalence relations in a given projective pointclass of even level assuming projective determinacy. These models are characterized by their correctness and the property that they correctly compute the tree from the appropriate scale. The main lemma shows that the tree from a scale can be reconstructed in a generic extension of an iterate of a mouse. We construct models with this property as generic extensions of iterates of mice if the corresponding projective ordinal is below ω2. On the way we consider several related problems, including the question when forcing does not add equivalence classes to thin projective equivalence relations. For example, we show that if every set has a sharp, then reasonable forcing does not add equivalence classes to thin provably ∆1 3 equivelence relations, and generalize this to all projective levels. 1.