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Model complete : ウィキペディア英語版
Model complete theory
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)
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