Large cardinals need not be large in HOD

Large cardinals need not be large in HOD

All Uncountable Regular Cardinals Can Be Inaccessible in Hod

Assuming the existence of a supercompact cardinal and an inaccessible above it, we construct a model of ZFC, in which all uncountable regular cardinals are inacces-

Collapsing the Cardinals of Hod

Assuming that GCH holds and κ is κ+3-supercompact, we construct a generic extension W of V in which κ remains strongly inaccessible and (α+)HOD < α+ for every infinite cardinal α < κ. In particular the rank-initial segment Wκ is a model of ZFC in which (α+)HOD < α+ for every infinite cardinal α.

Hod in Inner Models with Woodin Cardinals

We analyze the hereditarily ordinal definable sets HOD in the canonical inner model with nWoodin cardinals Mn(x, g) for a Turing cone of reals x, where g is generic over Mn(x) for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming Πn+2determinacy, for a Turing cone of reals x, HODn = Mn(M∞,Λ), whereM∞ is a direct limit of iterates of an initial segment ofMn+1 and Λ...

Ultrafilters and Large Cardinals

This paper is a survey of basic large cardinal notions, and applications of large cardinal ultrafilters in forcing. The main application presented is the consistent failure of the singular cardinals hypothesis. Other applications are mentioned that involve variants of Prikry forcing, over models of choice and models of determinacy. My talk at the Ultramath conference was about ultrafilters and ...