Measurable cardinals and the cardinality of Lindel\"of spaces

If it is consistent that there is a measurable cardinal, then it is consistent that all points g-delta Rothberger spaces have "small" cardinality.Comment: 9 pag

Dense ideals and cardinal arithmetic

From large cardinals we show the consistency of normal, fine, $\kappa$-complete $\lambda$-dense ideals on $\mathcal{P}_\kappa(\lambda)$ for successor $\kappa$. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman

Generic Large Cardinals and Systems of Filters

We introduce the notion of $\mathcal{C}$-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties.Comment: 36 page

Set Theory

This stimulating workshop exposed some of the most exciting recent develops in set theory, including major new results about the proper forcing axiom, stationary reflection, gaps in P(ω)/Fin, iterated forcing, the tree property, ideals and colouring numbers, as well as important new applications of set theory to C*-algebras, Ramsey theory, measure theory, representation theory, group theory and Banach spaces

Stationary set preserving L-forcings and the extender algebra

Wir konstruieren das Jensensche L-Forcing und nutzen dieses um die Pi_2 Konsequenzen der Theorie ZFC+BMM+"das nichtstationäre Ideal auf omega_1 ist abschüssig" zu studieren. Viele natürliche Konsequenzen der Theorie ZFC+MM folgen schon aus dieser schwächeren Theorie. Wir geben eine neue Charakterisierung des Axioms Dagger ("Alle Forcings welche stationäre Teilmengen von omega_1 bewahren sind semiproper") in dem wir eine Klasse von L-Forcings isolieren deren Semiproperness äquivalent zu Dagger ist. Wir verallgemeinern ein Resultat von Todorcevic: wir zeigen, dass Rado's Conjecture Dagger impliziert. Des weiteren studieren wir Generizitätsiterationen im Kontext einer messbaren Woodinzahl. Mit diesem Werkzeug erhalten wir eine Verallgemeinerung des Woodinschen Sigma^2_1 Absolutheitstheorems. We review the construction of Jensen's L-forcing which we apply to study the Pi_2 consequences of the theory ZFC + BMM + "the nonstationary ideal on omega_1 is precipitous". Many natural consequences ZFC + MM follow from this weaker theory. We give a new characterization of the axiom dagger ("All stationary set preserving forcings are semiproper") by isolating a class of stationary set preserving L-forcings whose semiproperness is equivalent to dagger. This characterization is used to generalize work of Todorcevic: we show that Rado's Conjecture implies dagger. Furthermore we study genericity iterations beginning with a measurable Woodin cardinal. We obtain a generalization of Woodin's Sigma^2_1 absoluteness theorem