Ultraproducts

Mathematical Logic · Axiom Academy

A powerful construction in model theory that builds new structures from infinite products via ultrafilters An ultrafilter on a set I is a collection U of subsets of I satisfying special properties. Think of it as a sophisticated way of defining "large" subsets. If A ∈ F and A ⊆ B ⊆ I , then B ∈ F There are two types of ultrafilters: Principal: U = A ⊆ I : i ∈ A for some fixed i ∈ I Non-principal (free): Contains no finite sets (only exists on infinite sets) Given a family of structures ( A i ) i∈I in the same language L , we form the product structure ∏ i∈I A i . Universe: ∏ i∈I | A i | = f : I → ⋃ i | A i | such that f ( i ) ∈ | A i | Functions: Applied coordinatewise: f ∏ ( a 1 ,..., a n )( i ) = f Aᵢ ( a 1 ( i ),..., a n ( i )) Relations: ( a 1 ,..., a n ) ∈ R ∏ iff ( a 1 ( i ),..., a n ( i )) ∈ R Aᵢ for all i ∈ I Each element of the product is a sequence ( a i ) i∈I where a i ∈ A i for each i . Operations and relations are evaluated "coordinatewise" across all indices. 3. Ultraproduct: Quotient by Ultrafilter The ultraproduct construction uses an ultrafilter to "quotient out" the product structure, identifying sequences that agree on a "large" set of indices. Equivalence relation: f ≡ U g iff i ∈ I : f ( i ) = g ( i ) ∈ U Ultraproduct: ∏ i∈I A i / U has universe (∏ i | A i |) / ≡ U Relations in ultraproduct: ( a 1 /U,..., a n /U) ∈ R iff i : ( a 1 ( i ),..., a n ( i )) ∈ R Aᵢ ∈ U When all A i = A are the same structure, we write A I / U and call it an ultrapower of A .

This is the written version of the interactive lesson above. See the full Mathematical Logic course.