Fragments of Martin's axiom and \(\Delta_ 3^ 1\) sets of reals
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.
scientific article; zbMATH DE number 682081
scientific article