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On ordinals accessible by infinitary languages

100%

Shelah S., Väisänen P., Väänänen J.

Fundamenta Mathematicae

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2005

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tom 186

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nr 3

193-214

EN

Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of $L_{λ⁺ω}$, with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with $⟨D^{ℳ },≺^{ℳ }⟩$ a well-ordering of type ≥ γ, then ϕ has a model ℳ ' where $⟨D^{ℳ '}, ≺^{ℳ '}⟩$ is non-well-ordered. One of the interesting properties of this number is that the Hanf number of $L_{λ⁺ω}$ is exactly $ℶ_{δ(λ)}$. It was proved in [BK71] that if ℵ₀ < λ < κ$ are regular cardinal numbers, then there is a forcing extension, preserving cofinalities, such that in the extension $2^{λ} = κ$ and δ(λ) < λ⁺⁺. We improve this result by proving the following: Suppose ℵ₀ < λ < θ ≤ κ are cardinal numbers such that ∙ $λ^{<λ} = λ$; ∙ cf(θ) ≥ λ⁺ and $μ^λ < θ$ whenever μ < θ; ∙ $κ^λ = κ$. Then there is a forcing extension preserving all cofinalities, adding no new sets of cardinality < λ, and such that in the extension $2^{λ} = κ$ and δ(λ) = θ.

2

A Cantor-Bendixson theorem for the space ${ω_1}^{ω_1}$

74%

Väänänen J.

Fundamenta Mathematicae

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1991

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tom 137

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nr 3

187-199

3

Δ-extension and Hanf-numbers

62%

Väänänen J.

Fundamenta Mathematicae

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1983

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tom 115

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nr 1

43-55

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Väänänen J.

1 Shelah S.

1 Väisänen P.

1 2005

1 1991

1 1983

On ordinals accessible by infinitary languages

A Cantor-Bendixson theorem for the space ${ω_1}^{ω_1}$

Δ-extension and Hanf-numbers