Given an uncountable cardinal κ with $\kappa ={\kappa }^{<\kappa }$ and $2^{\kappa }$ regular, we show that there is a forcing that preserves cofinalities less than or equal to $2^{\kappa }$ and forces the existence of a well-order of H(κ⁺) that is definable over ⟨H(κ⁺),∈⟩ by a Σ₁-formula with parameters. This shows that, in contrast to the case "κ = ω", the existence of a locally definable well-order of H(κ⁺) of low complexity is consistent with failures of the GCH at κ. We also show that the forcing mentioned above introduces a Bernstein subset...

Several results are presented concerning the existence or nonexistence, for a subset S of ω₁, of a real r which works as a robust code for S with respect to a given sequence $⟨S_{\alpha }:\alpha <\omega ₁⟩$ of pairwise disjoint stationary subsets of ω₁, where “robustness” of r as a code may either mean that $S\in L[r,⟨S{*}_{\alpha }:\alpha <\omega ₁⟩]$ whenever each $S{*}_{\alpha }$ is equal to $S_{\alpha }$ modulo nonstationary changes, or may have the weaker meaning that $S\in L[r,⟨S_{\alpha }\cap C:\alpha <\omega ₁⟩]$ for every club C ⊆ ω₁. Variants of the above theme are also considered which result when the requirement that S gets exactly...