An ultrapower is an important special case of the ultraproduct construction. Suppose κ is an infinite cardinal number and F is a structure in a firstorder theory; for example F could be a field. A filter U on κ is a subset of the power set P(κ) not containing the empty set and closed under finite intersections, and such that any superset of an element of U is also an element of U. The filter is an ultrafilter if for any member of P(κ) either it or its complement is in U, and is a free ultrafilter if it contains no finite sets (which entails it contains all cofinite sets.) An ultraproduct is a mathematical construction, which is used in abstract algebra to construct new fields from given ones, and in model theory, a branch of mathematical logic. ...
In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ...
In mathematics, a filter is a special subset of a partially ordered set. ...
In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ...
In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set. ...
An ultrapower of F is a quotient F^{κ}/U for some free ultrafilter on U, where we now may extend the constants, functions, and relations in F to the equivalence classes of F^{κ}/U. If c is a constant in F, then we have a corresponding constant in the ultrapower as the equivalence class of the element of F^{κ} which is the constant function with constant c. If δ < κ is an ordinal number, and f is an element of F^{κ}, we may call f_{δ}, the element of F which δ is mapped to, the component of f at δ. If we have a relation, we have a corresponding relation on the ultrapower which holds if the set of components on which it holds is in U. Functions defined on F are also defined on the ultrapower by applying F to each component. In mathematics, a quotient is the end result of a division problem. ...
Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...
