Ultraproduct

Ultraproduct

An ultraproduct is a mathematical construction, of which the ultrapower (defined below) is a special case. Ultraproducts are 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. One well-known 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.

The general method for getting ultraproducts uses an index set "I", a structure "M""i" for each element "i" of "I" (all of the same signature), and an ultrafilter "U" on "I". The usual choice is for "I" to be infinite and "U" to contain all cofinite subsets of "I". Otherwise the ultrafilter is principal, and the ultraproduct is isomorphic to one of the factors.

Algebraic operations on the Cartesian product

: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

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

and the ultraproduct is the quotient set with regard to ~. The ultraproduct is therefore sometimes denoted by

: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.

Other relations can be extended the same way: "a" "R" "b" if and only if

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

In particular, if every "M""i" is an ordered field, then so is the ultraproduct.

An ultrapower is an ultraproduct for which all the factors "M""i" are equal: M^kappa/U=prod_{alpha

Łoś's theorem

Łoś's theorem, also called "the fundamental theorem of ultraproducts", is due to Jerzy Łoś (the surname is pronounced, 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 "M""i" is a member of "U". More precisely:

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 phi

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 phi. The fact that 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.

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.

Analogously, one can define nonstandard complex numbers by taking the ultraproduct of copies of the field of complex numbers.

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.)

Ultralimit

:"For the ultraproduct of a sequence of metric spaces, see Ultralimit."In model theory and set theory, an ultralimit or limiting ultrapower is a direct limit of a sequence of ultrapowers.

Beginning with a structure, "A"0, and an ultrafilter, "D"0, form an ultrapower, "A"1. Then repeat the process to form "A"2, and so forth. For each "n" there is a canonical diagonal embedding A_n o A_{n+1}. At limit stages, such as "A"ω, form the direct limit of earlier stages. One may continue into the transfinite.

References

*
*


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • ultraproduct — noun A quotient of the direct product of a family of structures …   Wiktionary

  • metalogic — /met euh loj ik/, n. the logical analysis of the fundamental concepts of logic. [1835 45; META + LOGIC] * * * Study of the syntax and the semantics of formal languages and formal systems. It is related to, but does not include, the formal… …   Universalium

  • Ax–Kochen theorem — The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer d there is a finite set Yd of prime numbers, such that if p is any prime not in Yd then every homogeneous polynomial of degree d over the p adic… …   Wikipedia

  • Non-standard model of arithmetic — In mathematical logic, a nonstandard model of arithmetic is a model of (first order) Peano arithmetic that contains nonstandard numbers. The standard model of arithmetic consists of the set of standard natural numbers {0, 1, 2, …}. The elements… …   Wikipedia

  • Field (mathematics) — This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambiguation). In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it …   Wikipedia

  • Ultrafilter — In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged (as a filter). An ultrafilter may be considered as a finitely additive measure. Then every subset of X is …   Wikipedia

  • Compactness theorem — In mathematical logic, the compactness theorem states that a set of first order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for… …   Wikipedia

  • Jerzy Łoś — c.1955 Jerzy Łoś (born March 22, 1920 in Lwów (now Lviv, Ukraine) June 1, 1998 in Warsaw) (Polish pronunciation: [ˈjɛʐɨ ˈwɔɕ]) was a Polish mathematician, logician, economist, and philosopher. He is best known for his work on ultraproducts,… …   Wikipedia

  • Cartesian product — Cartesian square redirects here. For Cartesian squares in category theory, see Cartesian square (category theory). In mathematics, a Cartesian product (or product set) is a construction to build a new set out of a number of given sets. Each… …   Wikipedia

  • Reduction — Reduction, reduced, or reduce may refer to:cienceChemistry*Reduction – chemical reaction in which atoms have their oxidation number (oxidation state) changed. **Reduced gas – a gas with a low oxidation number **Ore reduction: see… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”