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

Peter HolyPhilipp Lücke — 2014

Fundamenta Mathematicae

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...

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