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In model theory, a branch of mathematical logic, a complete first-order theory ''T'' is called stable in λ (an infinite cardinal number), if the Stone space of every model of ''T'' of size ≤ λ has itself size ≤ λ. ''T'' is called a stable theory if there is no upper bound for the cardinals κ such that ''T'' is stable in κ. The stability spectrum of ''T'' is the class of all cardinals κ such that ''T'' is stable in κ. For countable theories there are only four possible stability spectra. The corresponding dividing lines are those for total transcendentality, superstability and stability. This result is due to Saharon Shelah, who also defined stability and superstability. == 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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stability spectrum」の詳細全文を読む スポンサード リンク
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