Martin's axioms, measurability and equiconsistency results

Journal of Symbolic Logic 54 (1):78-94 (1989)
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Abstract

We deal with the consistency strength of ZFC + variants of MA + suitable sets of reals are measurable (and/or Baire, and/or Ramsey). We improve the theorem of Harrington and Shelah [2] repairing the asymmetry between measure and category, obtaining also the same result for Ramsey. We then prove parallel theorems with weaker versions of Martin's axiom (MA(σ-centered), (MA(σ-linked)), MA(Γ + ℵ 0 ), MA(K)), getting Mahlo, inaccessible and weakly compact cardinals respectively. We prove that if there exists r ∈ R such that ω L[ r] 1 = ω 1 and MA holds, then there exists a ▵ 1 3 -selective filter on ω, and from the consistency of ZFC we build a model for ZFC + MA(I) + every ▵ 1 3 -set of reals is Lebesgue measurable, has the property of Baire and is Ramsey

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Citations of this work

Fragments of Martin's axiom and δ13 sets of reals.Joan Bagaria - 1994 - Annals of Pure and Applied Logic 69 (1):1-25.
Fragments of Martin's axiom and δ< sup> 1< sub> 3 sets of reals.Joan Bagaria - 1994 - Annals of Pure and Applied Logic 69 (1):1-25.
Projective forcing.Joan Bagaria & Roger Bosch - 1997 - Annals of Pure and Applied Logic 86 (3):237-266.
Exact equiconsistency results for Δ 3 1 -sets of reals.Haim Judah - 1992 - Archive for Mathematical Logic 32 (2):101-112.
Summable gaps.James Hirschorn - 2003 - Annals of Pure and Applied Logic 120 (1-3):1-63.

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References found in this work

Internal cohen extensions.D. A. Martin & R. M. Solovay - 1970 - Annals of Mathematical Logic 2 (2):143-178.
Some exact equiconsistency results in set theory.Leo Harrington & Saharon Shelah - 1985 - Notre Dame Journal of Formal Logic 26 (2):178-188.

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