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In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry. ==Definition== Fix a theory ''T'' with a model ''M''. The Morley rank of a formula φ defining a definable subset ''S'' of ''M'' is an ordinal or −1 or ∞, defined by first recursively defining what it means for a formula to have Morley rank at least α for some ordinal α. *The Morley rank is at least 0 if ''S'' is non-empty. *For α a successor ordinal, the Morley rank is at least α if in some elementary extension ''N'' of ''M'', ''S'' has countably many disjoint definable subsets ''Si'', each of rank at least ''α'' − 1. *For α a non-zero limit ordinal, the Morley rank is at least α if it is at least β for all β less than α. The Morley rank is then defined to be α if it is at least α but not at least ''α'' + 1, and is defined to be ∞ if it is at least α for all ordinals α, and is defined to be −1 if ''S'' is empty. For a subset of a model ''M'' defined by a formula φ the Morley rank is defined to be the Morley rank of φ in any ℵ0-saturated elementary extension of ''M''. In particular for ℵ0-saturated models the Morley rank of a subset is the Morley rank of any formula defining the subset. If φ defining ''S'' has rank α, and ''S'' breaks up into no more than ''n'' < ω subsets of rank α, then φ is said to have Morley degree ''n''. A formula defining a finite set has Morley rank 0. A formula with Morley rank 1 and Morley degree 1 is called strongly minimal. A strongly minimal structure is one where the trivial formula ''x'' = ''x'' is strongly minimal. Morley rank and strongly minimal structures are key tools in the proof of Morley's categoricity theorem and in the larger area of stability theory (model theory). 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「Morley rank」の詳細全文を読む スポンサード リンク
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