Cardinal transfer properties in extender models

Stationary Reflection in Extender Models

Working in L[E], we examine which large cardinal properties of κ imply that all stationary subsets of cof(<κ) ∩ κ reflect.

Determinacy in strong cardinal models

We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example: Theorem A Det(Π1-IND)⇒there exists an inner model with a strong cardinal. Theorem B Det(AQI)⇒there exist type-1 mice and hence inner models with proper classes of strong cardinals where Π1-IND (AQI) is the pointclass o...

A Characterization of (κ) in Extender Models

We prove that, in any fine structural extender model with Jensen’s λ-indexing, there is a (κ)-sequence if and only if there is a pair of stationary subsets of κ ∩ cof(< κ) without common reflection point of cofinality < κ which, in turn, is equivalent to the existence of a family of size < κ of stationary subsets of κ ∩ cof(< κ) without common reflection point of cofinality < κ. By a result of ...

Global square sequences in extender models

Rowbottom-type Properties and a Cardinal Arithmetic

Assuming Rowbottom-type properties, we estimate the size of certain families of closed disjoint functions. We show that whenever k is Rowbottom and 2W < HU1 (t), then 2<K = 2W or k is the strong limit cardinal. Next we notice that every strongly inaccessible Jónsson cardinal k is i/-Rowbottom for some v < k. In turn, Shelah's method allows us to construct a Jónsson model of cardinality k+ provi...