## Journal of Symbolic Logic

### Martin's Axioms, Measurability and Equiconsistency Results

#### 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 ($\mathrm{MA}(\sigma$-centered), ($\mathrm{MA}(\sigma$-linked)), $\mathrm{MA}(\Gamma^+_{\aleph_0}), \mathrm{MA}(K))$, getting Mahlo, inaccessible and weakly compact cardinals respectively. We prove that if there exists $r \in \mathbf{R}$ such that $\omega^{L\lbrack r\rbrack}_1 = \omega_1$ and MA holds, then there exists a $\triangle^1_3$-selective filter on $\omega$, and from the consistency of ZFC we build a model for $\mathrm{ZFC} + \mathrm{MA(I)} +$ every $\triangle^1_3$-set of reals is Lebesgue measurable, has the property of Baire and is Ramsey.

#### Article information

Source
J. Symbolic Logic, Volume 54, Issue 1 (1989), 78-94.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183742853

Mathematical Reviews number (MathSciNet)
MR987324

Zentralblatt MATH identifier
0673.03040

JSTOR