Numerical methods for ordinary differential equations

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. 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 non-standard analysis, which was pioneered (as an application of the compactness theorem) by Abraham Robinson.

Definition

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

{i \in I : a_{i} = b_{i}} \in U,

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

\prod_{i \in 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:

R ([a^{1}], \dots, [a^{n}]) ⟺ {i \in I : R^{M_{i}} (a_{i}^{1}, \dots, a_{i}^{n})} \in U,

where [a] denotes the equivalence class of a with respect to ~.

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^{κ} / U = \prod_{α < κ} M / U .

More generally, the construction above can be carried out whenever U is a filter on I; the resulting model $\prod_{i \in I} M_{i} / U$ 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 ω_i = 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 ψ_i = 2i. Because ψ_i > ω_i = i for all i, it follows that the equivalence class of ψ_i = 2i is greater than the equivalence class of ω_i = i, so that it can be interpreted as an infinite number which is greater than the one originally constructed. However, let χ_i = 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 Template:IPA-pl, 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 σ be a signature, $U$ an ultrafilter over a set $I$ , and for each $i \in I$ let $M_{i}$ be a σ-structure. Let $M$ be the ultraproduct of the $M_{i}$ with respect to $U$ , that is, $M = \prod_{i \in I} M_{i} / U .$ Then, for each $a^{1}, \dots, a^{n} \in \prod M_{i}$ , where $a^{k} = (a_{i}^{k})_{i \in I}$ , and for every σ-formula $ϕ$ ,

M ⊨ ϕ [[a^{1}], \dots, [a^{n}]] ⟺ {i \in I : M_{i} ⊨ ϕ [a_{i}^{1}, \dots, a_{i}^{n}]} \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 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 $S = {x \in M | R x}$ 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.

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, 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 $A_{n} \to A_{n + 1}$ . At limit stages, such as A_ω, form the direct limit of earlier stages. One may continue into the transfinite.

References

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Numerical methods for ordinary differential equations

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Definition

Examples

Łoś's theorem

Examples

Ultralimit

References

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Numerical methods for ordinary differential equations

Definition

Examples

Łoś's theorem

Examples

Ultralimit

References

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