Ultraproduct

An Ultra product is a construct in the field of model theory, a branch of mathematics. The objective of the design is to obtain a model (or many models ) for a given system of axioms another, which has unusual, in the language of the axiom system is not formalized properties. Idea of the construction is to define relations for consequences by a kind of majority decision.

Definition

Given any first-order language. Is an infinite index set, an ultra filter, which is not a main filter. Each is a model of language. On the Cartesian product

We define an equivalence relation by

And place on the set of equivalence classes following interpretation of the symbols of the language fixed: components be carried shortcuts; applies for each relation symbol

( Specifically, this is consistent with the definition of equality). Then the set of all equivalence classes modulo ~ of a model of the given language; it is called Ultra product.

Properties

Each formula in the language, which is met in each component, so all a given axiom system of first stage, as well as the Ultra product also applies to the Ultra product itself meet. For example, the ultra- product of bodies is a body, the ultra- small quantities of product, a minor amount, etc.

In contrast, this must for any statements that are not formalized, not true. Thus, as the induction axiom a statement about subsets ( and not elements ) of the set of natural numbers and not met in an ultra- product of infinitely many copies of the set of natural numbers.

The construction depends on; this leads in some cases to very specific set-theoretical questions in the theory of ultrafilters.

Ultra potencies

Often one chooses the same model for all and then receives a so -called Ultra power of this model. One example is the hyper real numbers. An analogous construction for the natural numbers results in a non-standard model of Peano arithmetic.

The embedding of a structure in their Ultra Potency is elementary.

Assuming the continuum hypothesis, one can show that certain ultra powers are isomorphic.