elementarily love

elementarily

Definitions

• adv. In an elementary way.

• adv. in an elementary manner

Etymologies

elementary +‎ -ly

Examples

• (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