Fragments of Martin's axiom and \(\Delta_ 3^ 1\) sets of reals
type: Item
instance of: scholarly article
title: Fragments of Martin's axiom and \(\Delta_ 3^ 1\) sets of reals
zbMATH Open document ID: 0810.03044
DOI: 10.1016/0168-0072(94)90017-5
author: Joan Bagaria
published in: Annals of Pure and Applied Logic
publication date: 1994-11-03T00:00:00Z
review text: The main result proved in the paper is the following statement: \(\text{MA}(\sigma\)-centered) + ``every Aronszajn tree is special'' + ``\(\omega_ 1\) is not inaccessible in \(L\)'' implies that there exists a \(\Delta_ 3^ 1\) set of reals without the property of Baire. The first part of the paper is devoted to an alternative proof of the related result by \textit{L. Harrington} and \textit{S. Shelah} [Notre Dame J. Formal Logic 26, 178-188 (1985; Zbl 0579.03039)] replacing Shelah's construction of a Cohen name for a Suslin tree by that of S. Todorčevič's. In the second part, the author determines the relative strength of several fragments of Martin's axiom, especially, he shows that MA(\(\sigma\)-centered) + ``every Aronszajn tree is special'' is actually weaker than Martin's axiom. The well readable paper is practically self-contained.
reviewed by: Lev Bukovský
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 682081
zbMATH Keywords: Martin's axiom
zbMATH Keywords: Baire property
zbMATH Keywords: \(\Delta_ 3^ 1\)-set
zbMATH Keywords: inaccessible cardinal
zbMATH Keywords: Aronszajn tree
Wikidata QID: Q114016506
MaRDI profile type: Publication
full work available at URL: 0168-0072(94)90017-5
OpenAlex ID: W1988339238
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description: scientific article; zbMATH DE number 682081