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In mathematical logic, Beth definability is a result that connects implicit definability of a property to its explicit definability, specifically the theorem states that the two senses of definability are equivalent. ==Statement== The theorem states that, given any two models ''A'' and ''B'' of a first-order theory ''T'' in the language L' ⊇ L such that ''A''|''L'' = ''B''|''L'' (where ''A''|''L'' is the reduct of ''A'' to ''L''), it is the case that ''A'' ⊨ φ() if and only if ''B'' ⊨ φ() (for φ a formula in L' and for all tuples a of ''A'') only if it is also the case that φ is equivalent modulo ''T'' to a formula ψ in ''L''. Less formally: a property is implicitly definable in a theory in language L (via introduction of a new symbol φ of an extended language L') only if that property is explicitly definable in that theory (by formula ψ in the original language L). Clearly the converse holds as well, so that we have an equivalence between implicit and explicit definability. That is, a "property" is implicitly definable with respect to a theory if and only if it is explicitly definable. The theorem does not hold if the condition is restricted to finite models. We may have ''A'' ⊨ φ() if and only if ''B'' ⊨ φ() for all pairs A,B of finite models without there being any ''L''-formula ψ equivalent to φ modulo T. The result was first proven by Evert Willem Beth. 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「Beth definability」の詳細全文を読む スポンサード リンク
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