Elementary equivalence

Two structures in the same first-order language are elementarily equivalent if they satisfy exactly the same first-order sentences. They may differ in size or internal structure despite sharing those sentences.

First-order logic: A formal logic whose formulas quantify over individual elements of a domain and use relations, functions, and predicates. Its sentences provide the tests that two structures must agree on.
Compactness theorem: A first-order theory has a model if every finite subset of its sentences has a model. It turns finite satisfiability into models that can share all first-order sentences.
Categorical theory: A theory with exactly one model up to isomorphism at a specified cardinality. Categoricity can force equivalent models of that size to be structurally identical.
Elementary substructure: A substructure that agrees with its larger structure on every formula using parameters from the substructure. The inclusion is elementary when it preserves more than just parameter-free sentences.
Löwenheim–Skolem theorem: A theorem giving models of first-order theories in multiple infinite cardinalities under broad conditions. It explains why sentence agreement often cannot determine a model's cardinality.
Structure (mathematical logic): A domain equipped with interpretations for the symbols of a formal language. Elementary equivalence compares structures interpreting the same symbols.
Back-and-forth method: A method that builds partial correspondences alternately in both directions to establish structural agreement. Back-and-forth constructions often prove that two structures satisfy the same sentences.
Dense linear order without endpoints: The theory of linear orders in which every interval contains another point and no first or last point exists. Its countable models are elementarily equivalent, though their underlying sets need not match.
Partial isomorphism: A bijection between subsets that preserves the structure's symbols on those subsets. Partial isomorphisms are local correspondences, unlike global sentence agreement.
Nonstandard analysis: A framework using nonstandard models of the real numbers to extend ordinary analysis. Elementary equivalence supports transferring first-order statements to nonstandard extensions.

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