Locally Σ₁-definable well-orders of H(κ⁺)

Peter Holy, and Philipp Lücke. "Locally Σ₁-definable well-orders of H(κ⁺)." Fundamenta Mathematicae 226.3 (2014): 221-236. <http://eudml.org/doc/282822>.

@article{PeterHoly2014,
abstract = {Given an uncountable cardinal κ with $κ = κ^\{<κ\}$ and $2^\{κ\}$ regular, we show that there is a forcing that preserves cofinalities less than or equal to $2^\{κ\}$ 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 $^\{κ\}κ$ that is definable over ⟨H(κ⁺),∈⟩ by a Δ₁-formula with parameters.},
author = {Peter Holy, Philipp Lücke},
journal = {Fundamenta Mathematicae},
keywords = {definable well-orders; failure of GCH; forcing},
language = {eng},
number = {3},
pages = {221-236},
title = {Locally Σ₁-definable well-orders of H(κ⁺)},
url = {http://eudml.org/doc/282822},
volume = {226},
year = {2014},
}

TY - JOUR
AU - Peter Holy
AU - Philipp Lücke
TI - Locally Σ₁-definable well-orders of H(κ⁺)
JO - Fundamenta Mathematicae
PY - 2014
VL - 226
IS - 3
SP - 221
EP - 236
AB - Given an uncountable cardinal κ with $κ = κ^{<κ}$ and $2^{κ}$ regular, we show that there is a forcing that preserves cofinalities less than or equal to $2^{κ}$ 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 $^{κ}κ$ that is definable over ⟨H(κ⁺),∈⟩ by a Δ₁-formula with parameters.
LA - eng
KW - definable well-orders; failure of GCH; forcing
UR - http://eudml.org/doc/282822
ER -