elementarily

• “(In the case of a complete theory, the models of different cardinality will be elementarily equivalent, even if they are non-isomorphic).”

  Mathematical logic and multiverses

• “Recalling that a complete theory T is one in which any sentence s, or its negation Not (s), belongs to the theory T, it follows that every model of a complete theory must be elementarily equivalent.”

  Mathematical logic and multiverses

• “Whilst isomorphic models must be elementarily equivalent, there is no need for elementarily equivalent models to be isomorphic.”

  Mathematical logic and multiverses

• “Two models of a theory are defined to be elementarily equivalent if they share the same truth-values for all the sentences of the language.”

  Mathematical logic and multiverses

• “The class of models of a complete theory will be mutually non-isomorphic, but they will nevertheless be elementarily equivalent.”

  Mathematical logic and multiverses

• “In this case, the models of the theory will be mutually non-isomorphic and elementarily inequivalent.”

  Mathematical logic and multiverses

• “Hence, mathematical logic suggests that the application of mathematical physics to the universe as a whole can generate two different types of multiverse: classes of non-isomorphic but elementarily equivalent models; and classes of non-isomorphic and elementarily inequivalent models.”

  Mathematical logic and multiverses

• “If two structures are isomorphic then they share all model-theoretic properties; in particular they are elementarily equivalent.”

  First-order Model Theory

• “This description implied among other things that the structures elementarily equivalent to R are exactly the real-closed fields, a class of fields which was already known to the algebraists in its own right.”

  First-order Model Theory

• “B are elementarily equivalent if and only if they have ultrapowers that are isomorphic to each other.”

  First-order Model Theory

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