Los's Theorem states that a first-order sentence holds in the ultraproduct of a family of structures, taken relative to some ultrafilter, if and only if the set of indices at which the sentence holds in the individual structures belongs to that ultrafilter. Named for Jerzy Los, it is the foundational result that makes the ultraproduct construction useful, underlying applications from nonstandard analysis to model-theoretic proofs of compactness.
Facts
Classification
Statement Form Statement FormCharacterization Theorem 1 StatementA first-order formula is true in the ultraproduct if and only if the set of indices where it is true in the individual structures belongs to the ultrafilter. 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Los's theorem (Wikipedia)
StatementQuote, Statement
any first-order formula is true in the ultraproduct if and only if the set of indices i such that the formula is true in M_i is a member of U.
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