By forcing over a model of 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 is a well-order of H definable over the structure H, 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. (shrink)

Given any subset A of ω1 there is a proper partial order which forces that the predicate xA and the predicate xω1A can be expressed by -provably incompatible Σ3 formulas over the structure Hω2,,NSω1. Also, if there is an inaccessible cardinal, then there is a proper partial order which forces the existence of a well-order of Hω2 definable over Hω2,,NSω1 by a provably antisymmetric Σ3 formula with two free variables. The proofs of these results involve a technique for manipulating the (...) guessing properties of club-sequences defined on stationary subsets of ω1 at will in such a way that the Σ3 theory of Hω2,,NSω1 with countable ordinals as parameters is forced to code a prescribed subset of ω1. On the other hand, using theorems due to Woodin it can be shown that, in the presence of sufficiently strong large cardinals, the above results are close to optimal from the point of view of the Levy hierarchy. (shrink)