The Uncountable Case
For a general stable theory T in a possibly uncountable language, the stability spectrum is determined by two cardinals κ and λ0, such that T is stable in λ exactly when λ ≥ λ0 and λμ = λ for all μ<κ. So λ0 is the smallest infinite cardinal for which T is stable. These invariants satisfy the inequalities
- κ ≤ |T|+
- κ ≤ λ0
- λ0 ≤ 2|T|
- If λ0 > |T|, then λ0 ≥ 2ω
When |T| is countable the 4 possibilities for its stability spectrum correspond to the following values of these cardinals:
- κ and λ0 are not defined: T is unstable.
- λ0 is 2ω, κ is ω1: T is stable but not superstable
- λ0 is 2ω, κ is ω: T is superstable but not ω-stable.
- λ0 is ω, κ is ω: T is totally transcendental (or ω-stable)
Read more about this topic: Stability Spectrum
Famous quotes containing the word case:
“A new talker will often call her caregiver “mommy,” which makes parents worry that the child is confused about who is who. She isn’t. This is a case of limited vocabulary rather than mixed-up identities. When a child has only one word for the female person who takes care of her, calling both of them “mommy” is understandable.”
—Amy Laura Dombro (20th century)