Elementary Equivalence

Mathematical Logic · Axiom Academy

Understanding when two structures satisfy exactly the same first-order sentences, and exploring the deep connections between elementary equivalence, isomorphism, and model theory Mathematical Logic • Unit 7 - Model Theory What is Elementary Equivalence? Two structures are elementarily equivalent if they cannot be distinguished by first-order logic—that is, they satisfy exactly the same first-order sentences. Formal Definition: Elementary Equivalence Let and be structures for the same language . We say are elementarily equivalent , written: if and only if for every sentence (closed formula) in : In words: Every sentence true in A is also true in B, and vice versa. Elementary Equivalence vs Isomorphism Elementary equivalence is a weaker notion than isomorphism. While isomorphic structures are always elementarily equivalent, the converse is not true. Theorem: Isomorphism Implies Elementary Equivalence If and are isomorphic, written , then: Proof idea: An isomorphism preserves all structure, including the truth of all formulas. If there exists a bijection that preserves all operations and relations, then both structures must satisfy the same sentences. Example: Non-Isomorphic but Elementarily Equivalent Structures The most famous example involves infinite structures that are elementarily equivalent but not isomorphic: the rationals and the reals as ordered fields. Classical Example: Dense Linear Orders Without Endpoints Consider the language of pure order (just the relation ).

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