The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory.
An ultraproduct is a quotient of the direct product of a family of structures.
The ultrapower is the special case of this construction in which all factors are equal.
Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was pioneered (as an application of the compactness theorem) by Abraham Robinson.
The general method for getting ultraproducts uses an index set
which compares components only relative to the ultrafilter
is an equivalence relation[proof 1] on the Cartesian product
was assumed to be an ultrafilter, the construction above can be carried out more generally whenever
The ultraproduct acts as a filter product space where elements are equal if they are equal only at the filtered components (non-filtered components are ignored under the equivalence).
One may define a finitely additive measure
Then two members of the Cartesian product are equivalent precisely if they are equal almost everywhere on the index set.
The ultraproduct is the set of equivalence classes thus generated.
Finitary operations on the Cartesian product
This constant map/tuple is an element of the Cartesian product
Analogously, one can define nonstandard integers, nonstandard complex numbers, etc., by taking the ultraproduct of copies of the corresponding structures.
As an example of the carrying over of relations into the ultraproduct, consider the sequence
so that it can be interpreted as an infinite number which is greater than the one originally constructed.
In the theory of large cardinals, a standard construction is to take the ultraproduct of the whole set-theoretic universe with respect to some carefully chosen ultrafilter
have a strong influence on (higher order) properties of the ultraproduct; for example, if
Łoś's theorem, also called the fundamental theorem of ultraproducts, is due to Jerzy Łoś (the surname is pronounced [ˈwɔɕ], approximately "wash").
It states that any first-order formula is true in the ultraproduct if and only if the set of indices
The theorem is proved by induction on the complexity of the formula
is an ultrafilter (and not just a filter) is used in the negation clause, and the axiom of choice is needed at the existential quantifier step.
As an application, one obtains the transfer theorem for hyperreal fields.
Since this can be translated into a first-order logical formula in the relevant formal language, Łoś's theorem implies that
Łoś's theorem does not apply to the Archimedean property, because the Archimedean property cannot be stated in first-order logic.
In model theory, this construction can be referred to as an ultralimit or limiting ultrapower.
form the direct limit of earlier stages.
The ultrafilter monad is the codensity monad of the inclusion of the category of finite sets into the category of all sets.
The codensity monad of the inclusion map