Ultraproduct   (Redirected from Łoś' theorem) Jump to navigation Jump to search 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.

Contents 1 Definition 2 Examples 3 Łoś's theorem 3.1 Examples 4 Direct limits of ultrapowers (ultralimits) 5 See also 6 References Definition The general method for getting ultraproducts uses an index set I, a structure Mi for each element i of I (all of the same signature), and an ultrafilter U on I. One usually considers this in the case that I to be infinite and U contains all cofinite subsets of I, i.e. U is not a principal ultrafilter. In the principal case the ultraproduct is isomorphic to one of the factors.

Algebraic operations on the Cartesian product {displaystyle prod _{iin I}M_{i}} are defined pointwise (for example, if {displaystyle +} is a binary function then {displaystyle a_{i}+b_{i}=(a+b)_{i}} ), and an equivalence relation is defined by {displaystyle asim b} if {displaystyle left{iin I:a_{i}=b_{i}right}in U,} and hence compares components only relative to the ultrafilter U. The ultraproduct is the quotient set with respect to {displaystyle sim .} The ultraproduct is therefore sometimes denoted by {displaystyle prod _{iin I}M_{i}/U.} and 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 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.

Other relations can be extended the same way: {displaystyle R([a^{1}],dots ,[a^{n}])iff left{iin I:R^{M_{i}}(a_{i}^{1},dots ,a_{i}^{n})right}in U,} where [a] denotes the equivalence class of a with respect to {displaystyle sim .} In particular, if every Mi is an ordered field, then so is the ultraproduct.

An ultrapower is an ultraproduct for which all the factors Mi are equal: {displaystyle M^{I}/U=prod _{iin I}M/U.,} More generally, the construction above can be carried out whenever U is a filter on I; the resulting model {displaystyle prod _{iin I}M_{i}/U} is then called a reduced product.

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. For example, the sequence ω given by ωi = i defines an equivalence class representing a hyperreal number that is greater than any real number.

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 ψ defined by ψi = 2i. Because ψi > ωi = i for all i, it follows that the equivalence class of ψi = 2i is greater than the equivalence class of ωi = i, so that it can be interpreted as an infinite number which is greater than the one originally constructed. However, let χi = i for i not equal to 7, but χ7 = 8. The set of indices on which ω and χ agree is a member of any ultrafilter (because ω and χ agree almost everywhere), so ω and χ belong to the same equivalence class.

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 U. Properties of this ultrafilter U have a strong influence on (higher order) properties of the ultraproduct; for example, if U is σ-complete, then the ultraproduct will again be well-founded. (See measurable cardinal for the prototypical example.) Łoś's theorem 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 i such that the formula is true in Mi is a member of U. More precisely: Let σ be a signature, {displaystyle U} an ultrafilter over a set {displaystyle I} , and for each {displaystyle iin I} let {displaystyle M_{i}} be a σ-structure. Let {displaystyle M} be the ultraproduct of the {displaystyle M_{i}} with respect to {displaystyle U} , that is, {displaystyle M=prod _{iin I}M_{i}/U.} Then, for each {displaystyle a^{1},ldots ,a^{n}in prod M_{i}} , where {displaystyle a^{k}=(a_{i}^{k})_{iin I}} , and for every σ-formula {displaystyle phi } , {displaystyle Mmodels phi [[a^{1}],ldots ,[a^{n}]]iff {iin I:M_{i}models phi [a_{i}^{1},ldots ,a_{i}^{n}]}in U.} The theorem is proved by induction on the complexity of the formula {displaystyle phi } . The fact that {displaystyle U} 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.

Examples Let R be a unary relation in the structure M, and form the ultrapower of M. Then the set {displaystyle S={xin M|Rx}} has an analog *S in the ultrapower, and first-order formulas involving S are also valid for *S. For example, let M be the reals, and let Rx hold if x is a rational number. Then in M we can say that for any pair of rationals x and y, there exists another number z such that z is not rational, and x < z < y. Since this can be translated into a first-order logical formula in the relevant formal language, Łoś's theorem implies that *S has the same property. That is, we can define a notion of the hyperrational numbers, which are a subset of the hyperreals, and they have the same first-order properties as the rationals. Consider, however, the Archimedean property of the reals, which states that there is no real number x such that x > 1, x > 1 + 1, x > 1 + 1 + 1, ... for every inequality in the infinite list. Łoś's theorem does not apply to the Archimedean property, because the Archimedean property cannot be stated in first-order logic. In fact, the Archimedean property is false for the hyperreals, as shown by the construction of the hyperreal number ω above.

Direct limits of ultrapowers (ultralimits) For the ultraproduct of a sequence of metric spaces, see Ultralimit.

In model theory and set theory, the direct limit of a sequence of ultrapowers is often considered. In model theory, this construction can be referred to as an ultralimit or limiting ultrapower.

Beginning with a structure, A0, and an ultrafilter, D0, form an ultrapower, A1. Then repeat the process to form A2, and so forth. For each n there is a canonical diagonal embedding {displaystyle A_{n}to A_{n+1}} . At limit stages, such as Aω, form the direct limit of earlier stages. One may continue into the transfinite.

See also Compactness theorem – Theorem Löwenheim–Skolem theorem – Existence and cardinality of models of logical theories Transfer principle – That all statements of some language that are true for some structure are true for another structure References Bell, John Lane; Slomson, Alan B. (2006) [1969]. Models and Ultraproducts: An Introduction (reprint of 1974 ed.). Dover Publications. ISBN 0-486-44979-3. Burris, Stanley N.; Sankappanavar, H.P. (2000) [1981]. A Course in Universal Algebra (Millennium ed.). show vte Mathematical logic Categories: Mathematical logicModel theoryNonstandard analysisTheorems in the foundations of mathematicsUniversal algebra

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