In this talk we introduce "self-genericity" axioms. Fixing an ideal I, we define the notion of "M is self-generic" (w.r.t I), where M is an elementary substructure of an initial segment of the universe, and consider several axioms asserting that these structures are frequent: Club Catch, Projective Catch and Stationary Catch (in decreasing order of strength). In particular, we show that Club Catch is equivalent to saturation. We also state some known consistency results related to these axioms, and note some connections with generic embeddings.
 

We will present two different ways to generically add a club subset of a successor cardinal, under some GCH.  The first one is designed to destroy a given stationary set, and we show that it also forces diamond.  The second adds a club with "small" conditions and destroys saturated ideals.  We will discuss the open problem of whether this can be done without any cardinal arithmetic assumptions.

This talk looks at the relationship between three foundational systems: Goedel's Constructible Universe of Sets, the naive conception of set found in consistent fragments of Frege's Grundgesetze, and the intensional logic of Church's Logic of Sense and Denotation. One basic result shows how to use the constructible sets to build models of fragments of Frege's Grundgesetze from which one can recover these very constructible sets using Frege's definition of membership. This result also allows us to solve the related consistency problem and joint consistency problems for abstraction principles with limited amounts of comprehension. Another basic aim of this paper is to show how to "factor'' this result via a consistent fragment of Church's Logic of Sense and Denotation: so one may use the constructible sets to build models of Church's Logic of Sense and Denotation, from which one may then define models of the consistent fragments of Frege's Grundgesetze.
Preprint: https://www.dropbox.com/s/afhcz8bzy4pdsoc/walsh-sean-CU%2BNC%2BIL-11-19-...
 

We continue with presentation of early applications of Martin's Maximum due to Foreman, Magidor and Shelah.

We will finish the proof that under GCH, dense ideals cannot exist at successors of singular cardinals.  Then we will outline how to separate the density property from the disjoint refinement property above aleph_1, and note remaining open questions.