Building new worlds from old ones
An ultrafilter votes on which sets of indices count as "most." Use that vote to blend many structures into one, and Łoś's theorem tells you exactly what is true in the blend.
Click subsets to mark them "large," or press a Principal button to fill one in. Watch the axiom checklist, and notice that every ultrafilter you manage to build is principal.
Build an ultrafilter on I = {1, 2, 3, 4, 5} by clicking subsets to mark them as "large." Or quick-fill a principal ultrafilter:
An ultrafilter U on an index set I is a maximal filter -- a collection of "large" subsets closed under supersets and finite intersections, satisfying the ultra property: for every A ⊆ I, either A ∈ U or I\A ∈ U. On a finite set, every ultrafilter is principal, generated by a single element. Non-principal ultrafilters on infinite sets exist by Zorn's lemma and are the key to ultraproduct constructions.
Key insight: An ultrafilter partitions all subsets into "large" and "small" with no ambiguity -- for every subset A, exactly one of A or its complement belongs to the ultrafilter. On a finite set this forces the ultrafilter to be principal, concentrating all "largeness" at a single point.
Choose the index the ultrafilter concentrates on and press Build Ultraproduct. Watch sequences that agree on a large set of indices merge into a single element.
Factor structures: M₁ = Z/2Z, M₂ = Z/3Z, M₃ = Z/5Z, M₄ = Z/7Z, M₅ = Z/11Z. Choose an ultrafilter and build the ultraproduct.
With finitely many factors, every ultrafilter is principal. A genuinely new ultraproduct needs infinitely many factors and a non-principal ultrafilter.
| Sequence | Z/2Z | Z/3Z | Z/5Z | Z/7Z | Z/11Z |
|---|---|---|---|---|---|
| zero | 0 | 0 | 0 | 0 | 0 |
| ones | 1 | 1 | 1 | 1 | 1 |
| p-1 | 1 | 2 | 4 | 6 | 10 |
| alternating | 0 | 1 | 0 | 1 | 0 |
| comp. alt. | 1 | 0 | 1 | 0 | 1 |
Given a family of structures (Mᵢ) indexed by I and an ultrafilter U on I, the ultraproduct ∏Mᵢ/U has as its domain the set of equivalence classes of sequences under the relation "agree on a set in U." Two sequences (aᵢ) and (bᵢ) represent the same element of the ultraproduct precisely when { i ∈ I : aᵢ = bᵢ } ∈ U.
Key insight: The ultraproduct "averages" the factor structures through the lens of the ultrafilter. A principal ultrafilter at index k simply recovers Mₖ, but a non-principal ultrafilter blends all factors together, producing a genuinely new structure that shares first-order properties with "most" of the factors.
Pick a sentence and an ultrafilter, then press Apply Łoś's Theorem. The sentence holds in the ultraproduct exactly when the set of factors where it is true is in the ultrafilter.
With finitely many factors, every ultrafilter is principal — non-principal ultrafilters only exist over infinite index sets.
Łoś's Theorem is the fundamental transfer principle: a first-order sentence holds in the ultraproduct if and only if the set of indices where it holds in the factors belongs to the ultrafilter. When all factors are the same structure M, the ultraproduct is an ultrapower, and the diagonal embedding M → Mᴵ/U is elementary.
Key insight: Łoś's Theorem is the engine behind ultraproduct arguments in model theory. It guarantees that truth transfers from the factors to the ultraproduct in a precise, formula-by-formula way. A sentence is true in ∏Mᵢ/U if and only if it is true in "U-almost all" of the factors.
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