On ordinals accessible by infinitary languages

• Autorzy: Saharon Shelah , Pauli Väisänen , Jouko Väänänen
• Języki publikacji: 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.

Kategorie tematyczne

• 03C75: Other infinitary logic
• 03E35: Consistency and independence results
• 03C55: Set-theoretic model theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2005
• Tom: 186
• Numer: 3
• Strony: 193-214

Daty

wydano

2005

Twórcy

autor

Saharon Shelah

• Institute of Mathematics, The Hebrew University, Jerusalem, Israel
• Department of Mathematics, Rutgers University, New Brunswick, NJ, U.S.A.

autor

Pauli Väisänen

• Department of Mathematics, University of Helsinki, P.O. Box 4, FIN-00014, Finland

autor

Jouko Väänänen

• Department of Mathematics, University of Helsinki, P.O. Box 4, FIN-00014, Finland

Identyfikatory

DOI

10.4064/fm186-3-1