Cardinals and iterations of HOD

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...

On the powersets of singular cardinals in HOD

From the assumption that there is a measurable cardinal κ with o(κ) = κ, we produce a model in which for all x ⊆ אω, HODx does not contain the powerset of אω. We also prove that this assertion requires large cardinals.

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-