Two structures M and N (same signature) are elementarily equivalent when they satisfy exactly the same first-order sentences (no parameters); symbolically M ≡ N. Equivalence concerns sentences, not necessarily formulas with parameters.

Principle

The organizing rule is agreement on the complete first-order theory in the given signature: M and N cannot be distinguished by any closed first-order sentence.

Demonstration

Example: The ordered sets (Q,<) and (R,<) are elementarily equivalent in the language of linear orders without endpoints because both satisfy the complete theory of dense linear orders without endpoints, despite being nonisomorphic.

Misapplication

Assuming elementary equivalence implies isomorphism or that it guarantees agreement on formulas with parameters; confusing M ≡ N with M ≺ N or with the existence of an embedding between them.

Consequence

Elementary equivalence means the models share the same first-order consequences and thus the same complete theory; this enables transfer of sentence-level properties and classification of models by theory rather than by isomorphism class.

Reversal

Non-equivalence is simply that there exists some first-order sentence true in one structure and false in the other, hence they realize different complete theories.

Boundary

Elementary equivalence is relative to a fixed signature and concerns sentences (no parameters). It does not imply elementarity of embeddings, agreement on parameterized formulas, nor preservation of model-theoretic invariants like cardinality or saturation.

Semantic Tension

Tension exists between elementary equivalence and structural similarity: two models may be elementarily equivalent yet very different (different cardinalities, topologies, or order types), highlighting the difference between syntactic theory and concrete structure.

Synthesis

Elementary equivalence collects structures that satisfy the same closed first-order sentences — they are indistinguishable at the level of the first-order theory even if they differ in size, topology, or other non-sentence features.