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In model theory, a first-order theory is called model complete if every embedding of models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula. This notion was introduced by Abraham Robinson. ==Model companion and model completion== A companion of a theory ''T'' is a theory ''T'' * such that every model of ''T'' can be embedded in a model of ''T'' * and vice versa. A model companion of a theory ''T'' is a companion of ''T'' that is model complete. Robinson proved that a theory has at most one model companion. A model completion for a theory ''T'' is a model companion ''T'' * such that for any model ''M'' of ''T'', the theory of ''T'' * together with the diagram of ''M'' is complete. Roughly speaking, this means every model of ''T'' is embeddable in a model of ''T'' * in a unique way. If ''T'' * is a model companion of ''T'' then the following conditions are equivalent: * ''T'' * is a model completion of ''T'' * ''T'' has the amalgamation property. If ''T'' also has universal axiomatization, both of the above are also equivalent to: * ''T'' * has elimination of quantifiers 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「Model complete theory」の詳細全文を読む スポンサード リンク
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