Large cardinals and locally defined well-orders of the universe

By forcing over a model of ZFC + GCH (above ℵ 0 ) with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H ( κ + ) ( κ ≥ ω 2 a regular cardinal) is a well-order of H ( κ + ) definable over the structure 〈 H ( κ + ) , ∈ 〉 by a parameter-free formula. Further, this forcing construction preserves all supercompact cardinals as well as all instances of regular local supercompactness. It is also possible to define variants of this construction which, in addition to forcing a locally defined well-order of the universe, preserve many of the n -huge cardinals from the ground model (for all n ).