On ordinals accessible by infinitary languages

Saharon Shelah, Pauli Väisänen, and Jouko Väänänen. "On ordinals accessible by infinitary languages." Fundamenta Mathematicae 186.3 (2005): 193-214. <http://eudml.org/doc/282975>.

@article{SaharonShelah2005,
abstract = {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.},
author = {Saharon Shelah, Pauli Väisänen, Jouko Väänänen},
journal = {Fundamenta Mathematicae},
keywords = {Hanf number; infinitary logic; accessible ordinal},
language = {eng},
number = {3},
pages = {193-214},
title = {On ordinals accessible by infinitary languages},
url = {http://eudml.org/doc/282975},
volume = {186},
year = {2005},
}

TY - JOUR
AU - Saharon Shelah
AU - Pauli Väisänen
AU - Jouko Väänänen
TI - On ordinals accessible by infinitary languages
JO - Fundamenta Mathematicae
PY - 2005
VL - 186
IS - 3
SP - 193
EP - 214
AB - 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.
LA - eng
KW - Hanf number; infinitary logic; accessible ordinal
UR - http://eudml.org/doc/282975
ER -