On ordinals accessible by infinitary languages

Saharon Shelah ; Pauli Väisänen ; Jouko Väänänen

Fundamenta Mathematicae, Tome 185 (2005), p. 193-214 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $⟨D^{ℳ ^{\prime }}, ≺^{ℳ ^{\prime }}⟩$ 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 $2λ = κ$ 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.

Détail

Publié le : 2005-01-01

Zbl 1096.03041

EUDML-ID : urn:eudml:doc:282975