Nonstandard Extension Research Articles

Nonstandard extension of quantum logic and Dirac's bra-ket formalism of quantum mechanics

An extension of the quantum logical approach to the axiomatization of quantum mechanics usingnonstandard analysis methods is proposed. The physical meaning of a quantum logic as a lattice of propositions is conserved by its nonstandard extension. But not only the usual Hubert space formalism of quantum mechanics can be derived from the nonstandard extended quantum logic. Also the Dirac bra-ket quantum mechanics can be derived as a consequence of such an extended quantum logic.

A nonstandard result about path continuity

The purpose of this note is to show that the nonstandard characterization of approximate continuity given in [4] by Wattenberg holds more generally in the context of "continuity paths" described by Bruckner, O'Malley and Thomson [3]. It is assumed that the reader is familiar with the principal elementary tools of nonstandard £nalysis: the transfer principle, internality and concurrence. [1] is a readily understandable reference. Beyond these, the more powerful extension of concurrence called saturation is of critical importance. [2] (page 27) illucidates this sufficiently. Let At denote a higher order structure including tt and its Lebesgue measure space, and *At a x-saturated nonstandard extension of At. With x assumed to be sufficiently large Wattenberg made the following definition. DEFINITION 1. x E *[0, 1] is negligible in case there is a standard set A C [0, 1] so that (i) st(x) is a point of dispersion for A, (ii) x 6 'A. Let Af denote the set of negligible points of *[0,1]. Then, Wattenberg proved the following:

Non-standard concave utility functions

Every reasonable concave preference ordering possesses a concave utility function assuming values in a suitable non-standard extension of the reals. Even if a real-valued concave utility function does exist, this function is not least concave if non-standard utilities are allowed, unless a certain finiteness (or piecewise linearity) condition holds.

Nonstandard nontransitive utility on mixture sets

We axiomatize three increasingly restrictive representations for a preference relation on a mixture set that presume neither transitivity nor an Archimedean condition. The representations involve a function from ordered pairs of elements in the mixture set into an ordered field extension of the reals. The axioms are formulated as finite cancellation conditions and are necessary and sufficient for the representations. The least general representation is a nonstandard extension of SSB (skew symmetric bilinear) utility.

On the cohomological dimension of non-standard number fields

The study made in this paper was motivated by the following question: given n 2 0, is the class of fields of cohomological dimension in elementary? It is relatively easy to see that this class is closed under elementary substructures, see Proposition 2.5; for n12 it is however unknown whether it is closed under ultraproducts, or even ultrapowers. Klingen showed in [6] that the class of fields having a class formation is elementary. As fields having a class formation have cohomological dimension 2, this suggests the following type of question: given an elementary class generated by fields of cohomological dimension 5 2, do all its elements have cohomological dimension 12? In this paper we propose to show that the elementary class S generated by all (totally imaginary) global fields and their algebraic extensions contains only fields of cohomological dimension ~2 (Theorem 3.6). This implies in particular that cd(Q*(i)) =2. We also study, for Q* a non-standard extension of Q, some of the properties of G(Q*); in contrast with the standard case, G(Q*) contains non-pro-cyclic abelian subgroups (Proposition 3.7), and cd(Q*(p))> 1 (Corollary 3.8). However, as in the standard case, we have cd(Q*“b)= 1 (Theorem 3.11). Our study allows us, given a prime number p, to find a sentence ulp such that for every field KES

Nonstandard definability

We investigate the notion of definability with respect to a full satisfaction class σ for a model M of Peano arithmetic. It is shown that the σ-definable subsets of M always include a class which provides a satisfaction definition for standard formulas. Such a class is necessarily proper, therefore there exist recursively saturated models with no full satisfaction classes. Nonstandard extensions of overspill and recursive saturation are utilized in developing a criterion for nonstandard definability. Finally, these techniques yield some information concerning the extendibility of full satisfaction classes from one model to another.

Monadic binary relations and the monad systems at near-standard points

AbstractLet (*X, *T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in *X is defined as m{x) = μT(st(x)). Consider the relationFrank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of Rns to the whole of *X. Wattenberg's extensions of Rns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of Rns to *X should be possible in any topological space. A study of such extensions of Rns is the purpose of the present paper. We call a binary relation W ⊆ *X × *X an infinitesimal on *X if it is monadic and reflexive on *X. We prove, among other things, that the existence of an infinitesimal on *X that extends Rns is equivalent to the condition that the space (X, T) be regular.

Monadic binary relations and the monad systems at near-standard points

AbstractLet (*X, *T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in *X is defined as m{x) = μT(st(x)). Consider the relationFrank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of Rns to the whole of *X. Wattenberg's extensions of Rns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of Rns to *X should be possible in any topological space. A study of such extensions of Rns is the purpose of the present paper. We call a binary relation W ⊆ *X × *X an infinitesimal on *X if it is monadic and reflexive on *X. We prove, among other things, that the existence of an infinitesimal on *X that extends Rns is equivalent to the condition that the space (X, T) be regular.

Generlized Fredholm transformations

AbstractIn an earlier paper we showed that the set ψ+ (X, Y) of super Tauberian transformations between two Banach spaces X and Y forms an open subset of B(X, Y) which is closed under perturbation by super weakly compact transformations. In this note we characterize a class dual to ψ+ (X, Y) which we denote by ψ-(X, Y). We show that T∈ψ+(X, Y) if and only if T′ ∈ ψ-(Y′, X′) and that T′∈ψ+(Y′, X′) if and only if T ∈ ψ-(X, Y) and provide standard and nonstandard characterizations of elements of ψ-(X, Y). These two classes thus play in some ways analogous roles to the sets of semi-Fredholm transforms ϕ+ (X, Y) and ϕ-(X, Y).Moreover en forms an open subset of B(X, Y) closed under the taking of adjoints, under the taking of nonstandard hull extensions, and under perturbation by super weakly compact transformations.

Nonstandard extensions of transformations between Banach spaces

Let X X and Y Y be (infinite-dimensional) Banach spaces and denote their nonstandard hulls with respect to an ℵ 1 \aleph _1 -saturated enlargement by X ^ \hat X and Y ^ \hat Y respectively. If B ( X , Y ) \mathcal {B}(X,Y) denotes the space of bounded linear transformations then a subset S S of elements of B ( X , Y ) \mathcal {B}(X,Y) extends naturally to a subset S ^ \hat S of B ( X ^ , Y ^ ) \mathcal {B}(\hat {X}, \hat {Y}) . This paper studies the behaviour of various kinds of transformations under this extension and introduces, in this context, the concepts of super weakly compact, super strictly singular and socially compact operators. It shows that B ( X , Y ) ^ ⫋ B ( X ^ , Y ^ ) \widehat {\mathcal {B}(X,Y)} \subsetneqq \mathcal {B}(\hat {X}, \hat {Y}) provided X X and Y Y are infinite dimensional and contrasts this with the inclusion K ( H ^ ) ⫋ K ( H ) ^ \mathcal {K}(\hat H) \subsetneqq \widehat {\mathcal {K}(H)} where K ( H ) \mathcal {K}(H) denotes the space of compact operators on a Hilbert space.

Weak Limits of Measures and the Standard Part Map

A construction is given, using the standard part map st, of the ${\text {weak}^\ast }$ standard part of an internal Baire measure in the nonstandard extension of a compact Hausdorff space. It is shown that the inverse image with respect to st of a Borel set is universally measurable with respect to completions of the $\sigma$-algebra generated by internal Baire sets. Applications and extensions of these results to noncompact spaces are given.

Analytic Sets, Baire Sets and the Standard Part Map

The problems considered here arose in connection with the interesting use by Loeb [8] and Anderson [1], [2] of Loeb's measure construction [7] to define measures on certain topological spaces. The original problem, from which the results given here developed, was to identify precisely the family of sets on which these measures are defined.To be precise, let be a set theoretical structure and * a nonstandard extension of , as in the usual framework for nonstandard analysis (see [10]). Let X be a Hausdorff space in and stx the standard part map for X, defined on the set of nearstandard points in *X. Suppose, for example, µ is an internal, finitely additive probability measure defined on the internal subsets of *X.

Weak limits of measures and the standard part map

A construction is given, using the standard part map st, of the weak ∗ {\text {weak}^\ast } standard part of an internal Baire measure in the nonstandard extension of a compact Hausdorff space. It is shown that the inverse image with respect to st of a Borel set is universally measurable with respect to completions of the σ \sigma -algebra generated by internal Baire sets. Applications and extensions of these results to noncompact spaces are given.

Nonstandard topological extensions: Addendum

It has recently come to my attention that, independent from my investigations in [1], Professor K.D. Stroyan [2] obtains for a Tychonoff space X and a normal base β a quotient space on *X which is homeomorphic to the Wallman-Frink Hausdorff compactification w(X, β). Consequently, if X is Tychonoff, then Corollary 4.1 in [1] implies that Stroyan's space is homeomorphic to the space w(X, β)S which is constructed in Theorem 4.1 in [1]. Stroyan's result is obtained by use of a monadic closure operator.I regret not mentioning Professor Stroyan's result in my paper [1].

Nonstandard topology and extensions of monad systems to infinite points

Intuitively, a topological space is a set together with some notion of “nearness”. Classically, this notion of nearness is usually described by a collection of open sets. With the development of Non-standard analysis by Abraham Robinson [11], we have an alternative way to describe this notion. If X is any set and *X is a nonstandard extension of X then we can describe a topology on X by means of a relationship of “infinitely close” on some points of X. In many ways this latter approach is more intuitive and leads to more straightforward proofs [11] than the classical approach. The first section of this paper explores the connections between these two approaches. In the main section of the paper we extend the notion of “infinitely close” to infinite points of *X. In the final section we give a very intuitive characterization of the compact-open topology (when the domain is locally compact) and use it to give quick, straight-forward proofs of the usual facts about this topology [1], [2]. This last section can be read independently of the second section.