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2 Fundamenta Mathematicae

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On partial orderings having precalibre-ℵ₁ and fragments of Martin's axiom

100%

Bagaria J. , Shelah S.

nr 2

181-197

We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-ℵ₁, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for σ-linked partial orderings. This yields a new solution to an old question of the first author about the relative strength of Martin's axiom for σ-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprāns and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-ℵ₁ property of a partial ordering while preserving its ccc-ness.

Generic absoluteness under projective forcing

100%

Bagaria J. , Bosch R.

Fundamenta Mathematicae

2007

tom 194

nr 2

95-120

We study the preservation of the property of LR being a Solovay model under projective ccc forcing extensions. We compute the exact consistency strength of the generic absoluteness of LR under forcing with projective ccc partial orderings and, as an application, we build models in which Martin's Axiom holds for $Σ\limits_{∼}}^{1}_{n}$ partial orderings, but it fails for the $Σ\limits_{∼}^{1}_{n+1}$.