Large cardinals and definable well-orders, without the GCH

Abstract

We show that there is a class-sized partial order P with the property that forcing with P preserves ZFC, supercompact cardinals, inaccessible cardinals and the value of 2κ for every inaccessible cardinal κ and, if κ is an inaccessible cardinal and A is an arbitrary subset of κκ, then there is a P-generic extension of the ground model V in which A is definable in 〈H(κ+)V[G],∈〉 by a Σ1-formula with parameters.We use this result to construct a class-sized partial order with the above preservation properties that forces the existence of well-orders of H(κ+) definable in the structure 〈H(κ+),∈〉 for every inaccessible cardinal κ. Assuming the GCH, David Asperó and Sy-David Friedman showed in [1] and [2] that there is a class-sized partial order preserving ZFC and various large cardinals and forcing the existence of a well-order of the universe whose restriction to H(κ+) is definable in 〈H(κ+)V[G],∈〉 by a parameter-free formula for every uncountable regular cardinal κ. Our second result can be interpreted as a boldface version of this result in the absence of the GCH.