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
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 Proof Method Classification
Statement Form Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Wolfram MathWorld: Los's Theorem -- from Wolfram MathWorld
Principal Result Of
Source Jerzy Łoś (Wikipedia)
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.
2. ProofWiki: Łoś's Theorem
ProofWikiProof sectionQuote, Proof section
We prove the $\LL$- sentences case by induction on the complexity of formulas. The general case trivially follows this proof.
Ultraproduct (Wikipedia)
In Group: Mathematical Logic and Model Theory, Lead or opening sectionQuote, In Group: Mathematical Logic and Model Theory, Lead or opening section
This mathematical construction appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory.
Jerzy Łoś (Wikipedia)
Principal Result Of: Jerzy Łoś, ContributionsQuote, Principal Result Of: Jerzy Łoś, Contributions
in particular for Łoś's theorem (also known as the Fundamental Theorem on Ultraproducts)
Wolfram MathWorld: Los's Theorem -- from Wolfram MathWorld
Wolfram MathWorldIn Branch: Logic and Foundations, Subject classificationsQuote, In Branch: Logic and Foundations, Subject classifications
Foundations of Mathematics Logic
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