We continue the exposition on self-genericity axioms for ideals on P(Z) (Club Catch, Projective Catch and Stationary Catch). We establish some relations with forcing axioms and with the existence of certain regular forcing embeddings, and also point out connections with Precipitousness. In particular we observe that if Projective Catch holds for an ideal, then that ideal is precipitous, and the converse holds for ideals that concentrate on countable sets. Finally we give an overview of the method used for proving the existence of models with Woodin cardinals coming from these axioms, using the Core Model Theory.

The tree property arises as the generalization of Koenig's infinity lemma to an uncountable cardinal.  The existence of an uncountable cardinal with the tree property has axiomatic strength beyond the axioms of ZFC.  Indeed a theorem of Mitchell shows that the theory ZFC + ``omega_2 has the tree property" is consistent if and only if the theory ZFC + ``There is a weakly compact cardinal" is consistent.  In the context of Mitchell's theorem, we can ask an old question in set theory:  Is it consistent that every regular cardinal greater than aleph_1 has the tree property? In this talk we will survey the best known partial results towards a positive answer to this question.
 

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-...