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

Examples

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Consider the language with one binary relation symbol '<'. The model R of real numbers and the model Q of rational numbers are elementarily equivalent, since they both translate '<' as an unbounded dense linear ordering.

There also exist non-standard models of number theory, which contain other objects than just the numbers 0, 1, 2, etc. However, the language is the same as standard number theory, since these extra objects cannot be mentioned. So, the standard model of number theory and these non-standard models are all elementarily equivalent. Model theory