On the separation of regularity properties of the reals

We present a model where omega (1) 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

Mathias absoluteness and the Ramsey property

In this article we give a forcing characterization for the Ramsey property of -Sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of real

Arboreal forcing notions and regularity properties of the real line

Die Arbeit befasst sich mit dem Studium von Regularitätseigenschaften der reellen Zahlen. Der Begriff Regularität wird, dank Verwendung von sogenannten arboreal forcings allgemein eingeführt. Insbesondere fokussieren wir uns auf die Frage der Trennung verschiedener Regularitätseigenschaften. Genauergesagt, falls P, Q arboreal forcings sind, konstruieren wir ein Modell indem sämtliche Teilmengen der reellen Zahlen P-messbar sind, aber zugleich eine Menge existiert, die nicht Q-messbar ist. Mit ähnlichen Mitteln werden auch Aussagen in der zweiten Ebene der projektiven Hierarchie untersucht. Schliesslich betrachten wir noch einige Fragen bezüglich Mass und Kategorie im verallgemeinerten Cantor Raum. Wir führen einen neuen Begriff für Mass in diesem Raum ein, der es uns erlaubt analoge Begriffe für Messbarkeit und überabzählbares random forcing zu entwickeln.The paper is centered around the study of regularity properties of the real line. The notion of regularity is presented in a rather general way, by using arboreal forcings. In particular, we focus on questions concerning the separation of different regularity properties. More precisely, in some cases, given P, Q arboreal forcings, we construct a model where all sets of reals are P-measurable and a non-Q-measurable set exists. A similar work is done for statements concerning the 2nd level of projective hierarchy. Finally, we also deal with questions about measure and category for the generalized Cantor space. In particular, we introduce a new notion of measure on such a space, which allows us to define the corresponding notion of measurability and the related uncountable random forcing