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Fragments of Martin's axiom and δ13 sets of reals

Joan Bagaria

Annals of Pure and Applied Logic 69 (1):1-25 (1994)

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A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees.Teruyuki Yorioka - 2010 - Annals of Pure and Applied Logic 161 (4):469-487.details We introduce a property of forcing notions, called the anti-, which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property . In this paper, we investigate the property . For example, we show that a forcing notion with the property does not add random reals. We prove that it is consistent that every forcing notion with the property has precaliber 1 and for forcing notions with the (...) Download Export citation Bookmark 5 citations
Summable gaps.James Hirschorn - 2003 - Annals of Pure and Applied Logic 120 (1-3):1-63.details It is proved, under Martin's Axiom, that all gaps in are indestructible in any forcing extension by a separable measure algebra. This naturally leads to a new type of gap, a summable gap. The results of these investigations have applications in Descriptive Set Theory. For example, it is shown that under Martin's Axiom the Baire categoricity of all Δ31 non-Δ31-complete sets of reals requires a weakly compact cardinal. Download Export citation Bookmark 1 citation
(2 other versions)1\ sets of reals.J. Bagaria & W. H. Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.details Download Export citation Bookmark 9 citations
Bounded forcing axioms and the continuum.David Asperó & Joan Bagaria - 2001 - Annals of Pure and Applied Logic 109 (3):179-203.details We show that bounded forcing axioms are consistent with the existence of -gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen's combinatorial principles for L at the level ω2, and therefore with the existence of an ω2-Suslin tree. We also show that the axiom we call BMM3 implies 21=2, as well as a stationary reflection principle which has many of the consequences of Martin's Maximum for objects of size 2. Finally, we give an (...) Download Export citation Bookmark 3 citations
Some weak fragments of Martin’s axiom related to the rectangle refining property.Teruyuki Yorioka - 2008 - Archive for Mathematical Logic 47 (1):79-90.details We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ℵ1 dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ℵ1 dense sets related to the rectangle refining property, and prove that they are really weaker fragments. Download Export citation Bookmark 4 citations
The relative strengths of fragments of Martin's axiom.Joan Bagaria - 2024 - Annals of Pure and Applied Logic 175 (1):103330.details Download Export citation Bookmark
Diamond principles in Cichoń’s diagram.Hiroaki Minami - 2005 - Archive for Mathematical Logic 44 (4):513-526.details We present several models which satisfy CH and some ♦-like principles while others fail, answering a question of Moore, Hrušák and Džamonja. Download Export citation Bookmark 2 citations
(2 other versions)Sets of reals.Joan Bagaria & W. Hugh Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.details Download Export citation Bookmark 5 citations
Projective forcing.Joan Bagaria & Roger Bosch - 1997 - Annals of Pure and Applied Logic 86 (3):237-266.details We study the projective posets and their properties as forcing notions. We also define Martin's axiom restricted to projective sets, MA, and show that this axiom is weaker than full Martin's axiom by proving the consistency of ZFC + ¬lCH + MA with “there exists a Suslin tree”, “there exists a non-strong gap”, “there exists an entangled set of reals” and “there exists κ < 20 such that 20 < 2k”. Download Export citation Bookmark 5 citations

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