We complete the exposition on self-genericity axioms for ideals on P(Z) (Club Catch, Projective Catch and Stationary Catch). We have established some relations with forcing axioms and with the existence of certain regular forcing embeddings and projections, and also point out connections with Precipitousness. We give an rough overview of the method used for proving the existence of models with Woodin cardinals coming from these axioms, using the Core Model Theory. In this talk we finish explaining the mechanism of absorbing extenders in the core model, and lifting iterability from countable models to models of large cardinality.
We complete the exposition on self-genericity axioms for ideals on P(Z) (Club Catch, Projective Catch and Stationary Catch). We have established some relations with forcing axioms and with the existence of certain regular forcing embeddings and projections, and also point out connections with Precipitousness. We give an rough overview of the method used for proving the existence of models with Woodin cardinals coming from these axioms, using the Core Model Theory. In this talk we explain the mechanism of absorbing extenders in the core model, and lifting iterability from countable models to models of large cardinality.
We continue the exposition on self-genericity axioms for ideals on P(Z) (Club Catch, Projective Catch and Stationary Catch). We have established some relations with forcing axioms and with the existence of certain regular forcing embeddings and projections, and also point out connections with Precipitousness. We give an rough overview of the method used for proving the existence of models with Woodin cardinals coming from these axioms, using the Core Model Theory. In this talk we explain the mechanism of absorbing extenders in the core model.
We continue the exposition on self-genericity axioms for ideals on P(Z) (Club Catch, Projective Catch and Stationary Catch). We have established some relations with forcing axioms and with the existence of certain regular forcing embeddings and projections, and also point out connections with Precipitousness. We give an rough overview of the method used for proving the existence of models with Woodin cardinals coming from these axioms, using the Core Model Theory. In this talk we explain one of the main technique used in the argument, namely the frequent extension argument.
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.
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.
Let \Gamma be a definable class of forcing posets and \kappa be a cardinal. We define MP(\kappa,\Gamma) to be the statement:
"For any A\subseteq \kappa, any formula \phi(v), for any P \in \Gamma, if there is a name \dot{Q} such that V^P models "\dot{Q}\in \Gamma + dot{Q} forces that \phi[A] is necessary" then V models \phi[A],"
where a poset Q \in \Gamma forces a statement \phi(x) to be necessary if for any \dot{R} such that V^Q \vDash \dot{R} \in \Gamma, then V^{Q\star \dot{R}} models \phi(x). When \Gamma is the class of proper forcings (or semi-proper forcings, or stationary set preserving forcings), we show that MP(\omega_1,\Gamma) is consistent relative to large cardinals. We also discuss the consistency strength of these principles as well as their relationship with forcing axioms. These are variants of maximality principles defined by Hamkins. This is joint work with Daisuke Ikegami.
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.
It is open whether \Pi^1_1 determinacy implies the existence of 0^\# in 3rd order arithmetic, call it Z_3. We compute the large cardinal strength of Z_3 plus "there is a real x such that every x-admissible is an L-cardinal." This is joint work with Yong Cheng.
