Given an uncountable cardinal κ with and regular, we show that there is a forcing that preserves cofinalities less than or equal to 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 of that is definable over ⟨H(κ⁺),∈⟩ by a Δ₁-formula with parameters.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-3-2,
author = {Peter Holy and Philipp L\"ucke},
title = {Locally S1-definable well-orders of H(k+)},
journal = {Fundamenta Mathematicae},
volume = {227},
year = {2014},
pages = {221-236},
zbl = {06314009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-3-2}
}
Peter Holy; Philipp Lücke. Locally Σ₁-definable well-orders of H(κ⁺). Fundamenta Mathematicae, Tome 227 (2014) pp. 221-236. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-3-2/