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• # Artykuł - szczegóły

## Fundamenta Mathematicae

2005 | 186 | 3 | 193-214

## On ordinals accessible by infinitary languages

EN

### Abstrakty

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.

193-214

wydano
2005

### Twórcy

autor
• Institute of Mathematics, The Hebrew University, Jerusalem, Israel
• Department of Mathematics, Rutgers University, New Brunswick, NJ, U.S.A.
autor
• Department of Mathematics, University of Helsinki, P.O. Box 4, FIN-00014, Finland
autor
• Department of Mathematics, University of Helsinki, P.O. Box 4, FIN-00014, Finland