Strong compactness and other cardinal sins

Indestructible Strong Compactness and Level by Level Equivalence with No Large Cardinal Restrictions

We construct a model for the level by level equivalence between strong compactness and supercompactness with an arbitrary large cardinal structure in which the least supercompact cardinal κ has its strong compactness indestructible under κ-directed closed forcing. This is in analogy to and generalizes [3, Theorem 1], but without the restriction that no cardinal is supercompact up to an inaccess...

Identity crises and strong compactness

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 note on strong compactness and resurrectibility

We construct a model containing a proper class of strongly compact cardinals in which no strongly compact cardinal κ is κ supercompact and in which every strongly compact cardinal has its strong compactness resurrectible.

On the Strong Equality between Supercompactness and Strong Compactness

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V |= ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V [G] |= ZFC + GCH in which, (a) (preservation) for κ ≤ λ regular, if V |...