Elementarily equivalent

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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