NationMaster - Encyclopedia: Ultraproduct

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 particular, it can be used in a "purely semantic" proof of the compactness theorem of first-order logic. Certainly the most important use of ultraproducts is the construction of the hyperreal numbers by taking the ultraproduct of countably infinitely many copies of the field of real numbers. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Jump to: navigation, search In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... The compactness theorem is a basic fact in symbolic logic and model theory and asserts that a set (possibly infinite) of first-order sentences is satisfiable, i. ... First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ... In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. ... Please refer to Real vs. ...

The general method for getting ultraproducts uses an index set I, a structure M_i for each element i of I, and an ultrafilter U on I (the usual choice is for I to be infinite and U to contain all cofinite subsets of I). 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 cofinite subset of a set X is a subset Y whose complement in X is a finite set. ...

Algebraic operations on the cartesian product In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ...

$prod_{i in I} M_i$

are defined in the usual way (for example, for a binary function +, (a + b) _i = a_i + b_i ), and an equivalence relation is defined by a ~ b if and only if In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

$left{ i in I: a_i = b_i right}in U,$

and the ultraproduct is the quotient set with regard to ~. The ultraproduct is therefore sometimes denoted by In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets...

$prod_{iin I}M_i / U .$

One may define a finitely additive measure m on the index set I by saying m(A) = 1 if A ∈ U and = 0 otherwise. 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. Measure can mean: To perform a measurement. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...

Other relations can be extended the same way: a R b if and only if In mathematics, an n-ary relation (or n-place relation or often simply relation) is a generalization of binary relations such as = and < which occur in statements such as 5 < 6 or 2 + 2 = 4. It is the fundamental notion in the relational model for databases. ...

$left{ i in I: a_i,R,b_i right}in U.$

In particular, if every F_i is an ordered field, then so is the ultraproduct. In mathematics, an ordered field is a field (F,+,*) together with a total order ≤ on F that is compatible with the algebraic operations in the following sense: if a ≤ b then a + c ≤ b + c if 0 ≤ a and 0 ≤ b then 0 ≤ a b It follows from these axioms...

Łoś' theorem

Łoś' theorem states that any first-order formula is true in the ultraproduct if and only if the set of indices i such that the formula is true in M_i is a member of U. More precisely: First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ...

Let $U$ be an ultrafilter over a set $I$ , and for each $i in I$ let $M i$ be a first order model. Let $M$ be the ultraproduct of the $M i$ with respect to $U$ , that is, $M = prod_{ iin I }M_i/U.$

Then, for each $[a^{1}], ldots, [a^{n}] in prod M_{i}$ , where $a^{k} = (a^{k}_{i})_{i in I}$ , and for every formula $φ$

$M models phi[[a^1], ldots, [a^n]]$ if and only if ${ i in I : M_{i} models phi[a^1_{i}, ldots, a^n_{i} ] } in U$ .

The theorem is proved by induction on the complexity of the formula $φ$ . The fact that $U$ is an ultrafilter (and not just a filter) is used in the negation clausule, and the axiom of choice is needed at the existential quantifier step.

Examples

The hyperreal numbers are the ultraproduct of one copy of the real numbers for every natural number, with regard to an ultrafilter over the natural numbers containing all cofinite sets. Their order is the extension of the order of the real numbers. In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. ... Please refer to Real vs. ...

Analogously, one can define nonstandard complex numbers by taking the ultraproduct of copies of the field of complex numbers. In the most restricted sense, nonstandard analysis or non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

Łoś' theorem GA_googleFillSlot("encyclopedia_square");

Examples

See also

Łoś' theorem