Cichoń's diagram, regularity properties and Δ13 sets of reals

Forcing axioms and projective sets of reals

This paper is an introduction to forcing axioms and large cardinals. Specifically, we shall discuss the large cardinal strength of forcing axioms in the presence of regularity properties for projective sets of reals. The new result shown in this paper says that ZFC + the bounded proper forcing axiom (BPFA) + “every projective set of reals is Lebesgue measurable” is equiconsistent with ZFC + “th...

On the separation of regularity properties of the reals

We present a model where ω1 is inaccessible by reals, Silver measurability holds for all sets but Miller and Lebesgue measurability fail for some sets. This contributes to a line of research started by Shelah in the 1980s and more recently continued by Schrittesser and Friedman (see [7]), regarding the separation of different notions of regularity properties of the real line. Acknowledgement. T...

Regularity Properties and Determinacy MSc

One of the most intriguing developments of modern set theory is the investigation of two-player infinite games of perfect information. Of course, it is clear that applied game theory, as any other branch of mathematics, can be modeled in set theory. But we are talking about the converse: the use of infinite games as a tool to study fundamental set theoretic questions. When such infinite games a...

Creature forcing and five cardinal characteristics in Cichoń's diagram

A General Setting for the Pointwise Investigation of Determinacy

It is well-known that if we assume a large class of sets of reals to be determined then we may conclude that all sets in this class have certain regularity properties: we say that determinacy implies regularity properties classwise. In [Lö05] the pointwise relation between determinacy and certain regularity properties (namely the Marczewski-Burstin algebra of arboreal forcing notions and a corr...