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2005 | 186 | 3 | 193-214
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On ordinals accessible by infinitary languages

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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.
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  • Institute of Mathematics, The Hebrew University, Jerusalem, Israel
  • Department of Mathematics, Rutgers University, New Brunswick, NJ, U.S.A.
  • Department of Mathematics, University of Helsinki, P.O. Box 4, FIN-00014, Finland
  • Department of Mathematics, University of Helsinki, P.O. Box 4, FIN-00014, Finland
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-3-1
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