The Stability Spectrum Theorem For Countable Theories
Theorem. Every countable complete first-order theory T falls into one of the following classes:
- T is stable in λ for all infinite cardinals λ. – T is totally transcendental.
- T is stable in λ exactly for all cardinals λ with λ ≥ 2ω. – T is superstable but not totally transcendental.
- T is stable in λ exactly for all cardinals λ that satisfy λ = λω. – T is stable but not superstable.
- T is not stable in any infinite cardinal λ. – T is unstable.
The condition on λ in the third case holds for cardinals of the form λ = κω, but not for cardinals λ of cofinality ω (because λ < λcof λ).
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