Ultraproduct

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. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal. For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this. 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 a structure (assumed to be non-empty in this article) for each element (all of the same signature), and an ultrafilter on For any two elements and of the Cartesian product declare them to be , written or if and only if the set of indices on which they agree is an element of in symbols, which compares components only relative to the ultrafilter This binary relation is an equivalence relation on the Cartesian product The is the quotient set of with respect to and is therefore sometimes denoted by or Explicitly, if the -equivalence class of an element is denoted by then the ultraproduct is the set of all -equivalence classes Although was assumed to be an ultrafilter, the construction above can be carried out more generally whenever is merely a filter on in which case the resulting quotient set is called a . When is a principal ultrafilter (which happens if and only if contains its kernel ) then the ultraproduct is isomorphic to one of the factors.

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MATH-381: Mathematical logic

Branche des mathématiques en lien avec le fondement des mathématiques et l'informatique théorique. Le cours est centré sur la logique du 1er ordre et l'articulation entre syntaxe et sémantique.

Ultrafilter on a set

In the mathematical field of set theory, an ultrafilter on a set is a maximal filter on the set In other words, it is a collection of subsets of that satisfies the definition of a filter on and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of that is also a filter. (In the above, by definition a filter on a set does not contain the empty set.) Equivalently, an ultrafilter on the set can also be characterized as a filter on with the property that for every subset of either or its complement belongs to the ultrafilter.

Ultrafilter

In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") is a certain subset of namely a maximal filter on that is, a proper filter on that cannot be enlarged to a bigger proper filter on If is an arbitrary set, its power set ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on are usually called . An ultrafilter on a set may be considered as a finitely additive measure on .

Real closed field

In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. A real closed field is a field F in which any of the following equivalent conditions is true: F is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in F if and only if it is true in the reals.