The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov’s 0-1 law that for any property which may or may not be satisfied by any given Turing degree, the satisfying class will either be of Lebesgue measure 0 or 1, so long as it is measurable. So either the typical degree satisfies the prop...

We begin the fine analysis of non Borel pointclasses. Working in ZFC + ̃ 11), we describe the Wadge hierarchy of the class of increasing differences of coanalytic subsets of the Baire space by extending results obtained by Louveau ([5]) for the Borel sets. Introduction. Collections of substets of the Baire space, the "logician’s reals", that are closed under continuous preimages have always been...

Given an arbitrary ideal f on the real numbers, two topologies are denned that are both finer than the ordinary topology. There are nonmeasurable, non-Baire sets that are open in all of these topologies, independent of f . This shows why the restriction to Baire sets is necessary in the usual definition of the ./"-density topology. It appears to be difficult to find such restrictions in the cas...