Elementarily equivalent

In mathematics, specifically model theory, two structures for a given language are said to be elementarily equivalent if any sentence satisfied by one model is also satisfied by the other.

Relationship to complete theories

If $T$ is a consistent theory, the following are equivalent:

* T is complete, meaning that for every sentence $phi$, $T\; vdash\; phi$ or $T\; vdash\; eg\; phi$,
* If $M$ and $N$ are models of $T$, then $M$ and $N$ are elementarily equivalent,
* $T$ has no consistent proper extension (it is a maximal consistent theory),
* There is a structure $M$ such that $T\; =\; Th(M)$, that is, such that for every sentence $phi$, $phi\; in\; T$ if and only if $M\; models\; phi$.

Examples

Consider the language with one binary relation symbol '<'. The model R of real numbers with its usual order and the model Q of rational numbers with its usual order are elementarily equivalent, since they both interpret '<' as an unbounded dense linear ordering, and since the theory of unbounded dense linear orderings is countably-categorical and does not have finite models, it is complete by Vaught's test.

There also exist non-standard models of number theory, which contain other objects than just the numbers 0, 1, 2, etc, and yet are elementarily equivalent to the standard model.

See also

* Elementary substructure

References

:"For further reading, see Model theory#References".
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