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

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Holy P., Lücke P.

Fundamenta Mathematicae

2014

tom 226

nr 3

221-236

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.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Holy P.

1 Lücke P.

1 2014

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