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Subjects
03-XX Mathematical logic and foundations
03Exx Set theory
03E47 Other notions of set-theoretic definability

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

Peter Holy, Philipp 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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