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ω-stable
In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify. Stable theories were the predominant subject of pure model theory from the 1970s through the 1990s, so their study shaped modern model theory and there is a rich framework and set of tools to analyze them. A major direction in model theory is "neostability theory," which tries to generalize the concepts of stability theory to broader contexts, such as simple and NIP theories. Motivation and history A common goa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stability Spectrum
In model theory, a branch of mathematical logic, a complete theory, complete first-order theory ''T'' is called stable in λ (an infinite cardinal number), if the Type (model theory)#Stone spaces, Stone space of every structure (mathematical logic), 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 line (model theory), dividing lines are those for totally transcendental theory, total transcendentality, superstable theory, superstability and stable theory, 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |