Reuniting the Antipodes - Constructive and Nonstandard Views of the Continuum
Symposion, 16 - 23 May 1999, San Servolo, Venice


Andreyev, Bell, Cutland, Dediu/Bridges, Giordano, Gordon/Andreev, Henle, Impens, Khalouani, Khan, Kopp, Lin, Laugwitz, J.R. Moschovakis, Di Nasso, Negri, Oikkonen, Perdry, von Plato, Render, Richman, Rosemeier, Ross, Schuster, Seisenberger, Spitters, Sun, Taschner, Terwijn, Veldman, Vernaeve, Waaldijk, Wolff, Wu, Zink,


Petr V. Andreyev (Nizhnii Novgorod State University, Russia)
Definable standardness predicates in Internal Set Theory

We consider unary predicates defined by a formula which extend the standardness predicate in Internal Set Theory. We call them definable standardness predicates if they expose some features analogous to those of original standardness predicate.

The main result is that under some natural restrictions the only definable predicate obeying all the three principles of Idealization, Standardization and Transfer is the 'usual' standardness predicate.
We also give two interesting examples of definable standardness predicates.

John L. Bell (University of Western Ontario, Canada)
The continuum in smooth infinitesimal analysis

In models of smooth infinitesimal analysis (SIA), all maps on the continuum are infinitesimally affine, hence arbitrarily many times differentiable, and the continuum itself is indecomposable, that is, not the disjoint union of two nonempty parts. It follows that the law of excluded middle must fail in models of SIA, and reasoning therein must be constructive. I shall enlarge on properties of the continuum in such models, pointing out the contrasts with nonstandard analysis. Time permitting, I shall also sketch how the differential calculus is developed in such models.

Nigel J. Cutland (University of Hull, England)
Constructive aspects of nonstandard methods in fluid mechanics

Classical proofs of existence of solutions to the Navier-Stokes equations in three dimensions are highly non-constructive, and the question of uniqueness is a major open problem. This raises the question of what could be a valid constructive approach to these equations that could provide a theoretical underpinning for the kind of numerical approach employed in real-world calculations.

On the other hand, a nonstandard approach utilises uniqueness in a hyperfinite approximation to the equations, so there is the usual constructiveness in this sense at the nonstandard level, from which solutions are obtained by taking standard parts as usual. We will illustrate these ideas in both the deterministic and stochastic settings from recent work with Marek Capinski.

Luminita S. Dediu (University of Canterbury, Christchurch, New Zealand)
Embedding a linear subset of B(H) in its predual

Let $H$ be a Hilbert space and let ${\cal R}$ be a linear subset of the space ${\cal B}(H)$ of all bounded linear operators on $H$. Denote by ${\cal R}_1$ the unit ball of ${\cal R}$, and by ${\cal R}_{\#}$ the space of all linear functionals on ${\cal R}$ that are weak--operator continuous on ${\cal R}_1$. We prove that if ${\cal R}_1$ is weak--operator totally bounded, then there exists a one--one linear mapping $\Phi$ of ${\cal R}$ into ${\cal R}_{\#}^{*}$ -- the dual space of ${\cal R}_{\#}$-- such that

$\Phi$ is uniformly continuous on ${\cal R}_1$

$\phi({\cal R}_1)$ is dense in the unit ball of ${\cal R}_{\#}^{*}$

the restriction of $\Phi^{-1}$ to $\Phi({\cal R}_1)$ is uniformly continuous.

Paolo Giordano (Università di Novara, Italy)
Nilpotent infinitesimals and synthetic differential geometry in classical logic

An extention of *R with nilpotent infinitesimals (e.g. $h\ne 0$ but $h^2=0$) is presented in order to obtain results similar to Kock--Lawvere's synthetic differential geometry [Kock, 1981], but in a classical and not intuitionistic context and only with elementary category theory without any use of Topos theory. We show that with a little modification of Kock--Lawvere's axiom is possible to construct a simple ring containing $^*\R$ and nilpotent numbers, but without the usual incompatibility with classical logic. The same idea of extension can be used to add infinitesimal points to spaces similar to Chen's one [Chen, 1982]. Many results of [Kock, 1981] can be likewise repeated in this context without the necessity of using intuitionistic logic; unfortunately in the category of extended spaces the ``cartesian lifting'' works only for some spaces, and we give a useful sufficient condition for this.

P.V.Andreyev and Evgenii I. Gordon (Nizhnii Novgorod State University, Russia)
Mathematics in Hyperfinite Sets Theory

The Nonstandard Class Theory (NCT) was introduced in [Go] (see also [AG] for its modified version). This theory is formulated on the base of Von Neumann - Gödel - Bernays Theory in the same way as Nelson's IST - on the Zermelo Fraenkel's Set Theory. The Hyperfinite Sets Theory (HST) is obtained from NCT by replacment of Infinity Axiom by its negation. The family of subclasses of $V_\omega$ in any model of NCT forms a model of HST and thus HST is consistent. We consider the hyperfinite mathematics, which can be formulate in HST. For example, we formulate there some hyperfinte version of Commutative Harmonic Analysis, in particular a hyperfinite version of Pontrjagin's duality theorem. On the base of Hyperfinte Harmonic Analysis the Harmonic Analysis on Locally Compact Abelian Groups can be developed [Go]

[AG] P.V.Andreev and E.I.Gordon. Nonstandard Class Theory. Bull. Symbolic Logic, vol. 5, no. 1 (1999), p. 88 -- 89

[Go] E.I.Gordon. Nonstandard Mehods in Commutative Harmonic Analysis. AMS. Providence, Rhode Island, 1997.

James Henle (Smith College, Northhampton, Massachusetts)
Non-nonstandard analysis: category, measure, and integration

Before nonstandard analysis, Detlef Laugwitz and Curt Schmieden developed a system of analysis that included infinitesimals. We build an extension of this system to prove the Baire Category Theorem and to construct Lebesgue measure. Extensions can be formed that are, in some sense, enlargements, that are $\kappa$-saturated for a given $\kappa$, and that can produce Loeb-like measures.

Chris Impens (University of Gent, Belgium)
Some thoughts on nonstandard geometry

A plane geometry is considered whose straight lines are copies not of the set of reals (as in Hilbert's axiomatized geometry) but of the set of hyperreals. This non-archimedean geometry, possessing the standard part mapping and thus avoiding the flaws imputed on Veronese's system, might lead to the view that Euclidean geometry is the standard part of some nonstandard absolute geometry (i.e. without parallel axiom). To this end, some basic theorems of absolute geometry are revisited in a non-archimedean framework.

Mohamed Khalouani (Université de Franche-Comté, Besançon, France)
Étude constructive de problemes de topologie pour les reels irrationnels

We study in a constructive manner some problems of topology related to the set Irr of irationals reals.

We show that the set Irr is one-to-one with the set Dfc of infinite developements in continued fraction (dfc). We define two extensions of Irr Dfc_1 and Dfc_2.

We introduce six natural distances over Irr wich we denote by dfc_0, dfc_1, dfc_2, d, d_mir and d_cut. We show that only the four distances dfc_0, dfc_1, d and d_mir among the six make Irr a complete metric space. The last distances define in Irr the same topology in a constructive sens.

We study further the set Dfc_1 in wich, we show notably that the irrationals constiyue a closed subset. Finally, we make a particular study of the completion of Dfc for the two equivalent metrics dfc_2 and d_cut.

M. Ali Khan
Modelling `negligibility' in mathematical economics: an application of Loeb spaces

We discuss how a particular class of standard measure spaces, hyperfinite Loeb spaces, can be used to model situations where individual players are strategically negligible, as in large 'non-anonymous' games, or where information is diffused, as in games with imperfect information. We present results on the existence of Nash equilibria in both kinds of games. Our results cover the case when the action sets are taken to be the unit interval, results now known to be false when they are based on more familiar measure spaces such as the Lebesgue unit interval. We also emphasize three criteria for the modelling of such game-theoretic situations -- asymptotic implementability, homogeneity and measurability -- and argue for games on hyperfinite Loeb spaces on the basis of these criteria.

P. Ekkehard Kopp (University of Hull, England)
Hyperfinite discretisations and convergence in option pricing models

Questions of convergence of financial option pricing models remain a topic of much debate in mathematical finance. The use of hyperfinite discrete models allows a strenghtening of the usual weak convergence results which leads to results on the convergence of trading strategies as well as option prices for the standard Black-Scholes-type models. Recent work shows that aspects of this theory can be extended to incomplete market models, where they confirm the robustness of the Foellmer-Schweizer minimal martingale measure approach under convergence from discrete to continuous-time models.

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Detlef Laugwitz (Technische Universität Darmstadt, Germany)
Curt Schmieden's approach to infinitesimals---an eye-opener to the historiography of analysis

In the early 1950's C. Schmieden developed the idea to 'adjoin' an infinitely large natural number Omega to real analysis. A formula $AOmega$ was true if $A(n)$ was true for almost all finite natural $n$. In a paper of 1958 we used rational (and later real) sequences to establish a model in which Omega was represented by the sequence of the finite natural numbers. Clearly, this approach was less powerful with respect to new applications than Robinson's was. Yet it turned out that the Omega Calculus could serve as a means to clarify earlier uses of infinitesimals and infinitely large numbers. The sequential approach was successful in re- considering the foundational aspects of Cauchy's textbooks, and also early Fourier analysis and the use of delta functions around 1820. Adjoining an ideal element was more in the spirit of Euler (and, in some sense, of Leibniz and Bolzano). Like all of pre-Cantorian mathematics the two versions of Schmieden's approach were basically constructive.

Xiaoai Lin (National University of Singapore)
On the almost independence of correspondences on Loeb space

A correspondence is a mapping whose values are nonempty sets. It is also called a set-valued mapping or multifunction. The study of measurable selections of correspondences traces its history back to the work of Von Neumann in 1930s. This study has led to wide applications in stochastic analysis, control theory, optimization and mathematical economics in recent years.

A purpose of this paper is to study correspondences and their measurable selections in the context of stochastic independence. This work is partly motivated by the mathematical modeling of large economic systems with individual uncertainty where the optimal choices of economic agents may not be unique. In particular, we show that an almost independent correspondence can always be represented as the closure of a sequence of its measurable selections which are almost independent themselves. We also characterize the condition of almost independence for correspondences via inverse images of other type of sets other than open sets.

Joan R. Moschovakis (Occidental College, Los Angeles) (University of Athens, Greece)
The intuitionistic continuum as an extension of the classical one

From the classical set-theoretic point of view, Brouwer's ``sharp arrows'' are definable sequences, and his celebrated claim that ``Every completely defined function is continuous'' depends on his acceptance of incompletely defined elements of Baire space.

Using the consistent (and classically and constructively plausible) assumption that every $\Delta^2_1$ well ordering of a subclass of Baire space is countable, one can build a (nonconstructive) realizability model of a theory which presents the intuitionistic continuum as an extension of the classical ``lawlike'' continuum. Within the intuitionistic continuum, a dense definable class of new ``lawless'' (or generic) sequences obeys suitably restricted principles of open and closed data. The Kleene-Troelstra Principle, an extension of Brouwer's Principle, holds; in particular, every function defined on all reals, all lawless reals, or all hesitant (not not lawlike) reals is continuous on its domain.

Mauro Di Nasso(Università di Pisa, Italy)
Hyperreals as Ideal Limits: an Elementary Axiomatic Presentation of Nonstandard Analysis

We axiomatize a notion of limit value for N-sequences, by considering an ideal natural number $\alpha$. The behaviour of limit values is ruled by the following five properties. 1) If $c_\emptyset=\langle\emptyset:n\inN\rangle$ is the $\emptyset$-constant sequence, then its ideal value $c_\emptyset(\alpha)=\emptyset$.
2) If $1_N:N\toN$ is the identity map, then $1_N(\alpha)=\alpha$.
3) $f(\alpha)\in g(\alpha)$ iff $f(\alpha)=h(\alpha)$ for a sequence $h$ with $h(n)\in g(n)$ for all $n\inN$.
4) If $f(n)=\{g(n),h(n)\}$ for all $n\inN$, then $f(\alpha)=\{g(\alpha),h(\alpha)\}$.
5) If $f(\alpha)=g(\alpha)$, then $(\varphi\circ f)(\alpha)=(\varphi\circ g)(\alpha)$ for every function $\varphi$ (provided compositions $\varphi\circ f$ and $\varphi\circ g$ are defined).
All axioms of Zermelo-Fraenkel set theory with choice (without regularity) are also postulated. A nonstandard embedding $*:V\to V$ is defined on the universal class of all sets, by setting $*A=c_A(\alpha)$, where $c_A$ is the $A$-constant sequence. Thus the hyperreal number system $*R$ is the set of all ideal values of sequences of real numbers. The basic principles of nonstandard analysis, namely transfer and countable saturation, are proved. The resulting nonstandard set theory is a conservative extension of ZFC.

Sara Negri (University of Helsinki, Finland)
On the proof theory of constructive ordered fields

As a first stage for a proof theory of the constructive continuum, a sequent calculus system for the theory of constructive ordered fields is presented. An axiomatization of ordered fields is given, based on a single primitive relation of constructive linear order. The axioms are converted into sequent calculus nonlogical rules of inference. The special form of the nonlogical rules is such that the structural rules of weakening, contraction and cut can be proved admissible also in this extension of the logical sequent calculus.

Juha Oikkonen (University of Helsinki, Finland)
Some geometric ideas related to Brownian motion

We schall discuss some geometric ideas about the continuum, the plane and more generally about Riemannian manifolds in connection to nonstandard constructions of Brownian motions in these spaces.

Hervé Perdry (Université de Franche-Comté, Besançon, France)
Computing in the constructive henselisation of a valued field

In a previous paper (Construction du hensélisé d'un corps valué, Prépublications de l'Équipe de Mathématiques de Besançon), Franz--Victor Kuhlmann and Henri Lombardi gave a construction of the Henselisation $K^h$ of a valued field $K$. We give here some algorithms for more computations in $K^h$, and a construction of $K^{sh}$, the strict Henselisation of $K$.

Jan von Plato (University of Helsinki, Finland)
The lattice of open intervals of the intuitionistic continuum

Due to the undecidability of order on reals, there is no simple uniformly applicable operation of meet of two open intervals that would always produce non-negative intervals as values. A lattice structure, constructively stronger than the usual one, is given for open intervals, and the problem of meet solved by a suitable definition of apartness of intervals. A constructive counterpart to the equational characterization of partial order is given in terms of apartness.

Herrmann Render (Universität Duisburg, Germany)
Borel measure extensions of measures defined on sub-$\sigma$-algebras

We develop a new approach to the measure extension problem, based on Nonstandard Analysis. The class of thick topological spaces, which includes all locally compact and all K-analytic spaces, is introduced in this paper, and measure extension results of the following type are obtained: If $(X,{\cal T})$ is a regular, Lindelöf and thick space, ${\cal A} \subset \sigma [{\cal T}]$ is a $\sigma$-algebra, and $\nu$ is a finite measure on ${\cal A }$, inner regular with respect to the closed sets in ${\cal A }$, then $\nu$ has a Radon extension. The methods developed here allow us to improve on previously known extension results.

Fred Richman (Florida Atlantic University, Boca Raton)
Constructive mathematics without choice

The countable axiom of choice is accepted in most versions of constructive mathematics. One consequence (or cause) of this is the predominance of sequences in constructive practice. It is argued here that the mathematics becomes better if one rejects this axiom. Two case studies are considered: the fundamental theorem of algebra, and the theory of trace-class operators on a Hilbert space.

Frank Rosemeier (Fernuniversität Hagen, Germany)
On Conway-numbers and generalized real numbers

John H. Conway describes in his book ``On Numbers and Games'' (ONAG) a general method to create a class of numbers containing all real numbers as well as every ordinal number. Using the logical law of the excluded middle (LEM) he equips this class with the structure of a totally ordered field. This talk is a first step to investigate the contribution of Conways theory to the foundations of Constructive Nonstandard Analysis. In ONAG Conway suggests to define real numbers as (Conway-)cuts in the set of rational numbers. Following this idea, a constructive notion of real numbers will be developed and parallels to, and differences from, the concept of generalized real numbers recently published by Fred Richman (Indag. Mathem., N.S., 9 (4), 595-606) will be investigated.

David Ross (University of Hawaii)
The constructive content of nonstandard measure existence proofs: is there any?

It is common to say of nonstandard proofs that they are "constructive modulo an ultrafilter." For example, the starting point for an existence argument using Loeb measures will often be a finite measure produced using purely combinatorial methods. In this talk I look at some nonstandard measure existence arguments, show how they might suggest constructive arguments, but also identify some serious obstacles to their constructive adaptation.

Monika Seisenberger (Universität München, Germany)
Kruskal's tree theorem in a constructive theory of inductive definitions

Traditionally Kruskal's tree theorem is formulated by use of well quasi orders:

If (A,<=) is a well quasi order, then T(A), the set of finite trees with labels in A, together with the embedding relation is a well quasi order, as well.

Hereby a set A with a reflexive and transitive relation <= is a well quasi order iff there is no infinite bad sequence in A, i.e. if for every infinite sequence (a_i)_{i < omega} there exists indices i < j < omega such that a_i <= a_j.

We give a constructive proof of Kruskal's theorem by using an inductive formulation of being a well quasi order.

Peter Schuster (Universität München, Germany)
Elementary choiceless constructive analysis

Elementary constructive analysis can well be treated with rather little countable choice, provided that Cauchy sequences as basic notions are replaced with Dedekind cuts, or formal reals. We illustrate this by examples concerning, for instance, completeness and indecomposability of the continuum, least upper bounds, and intermediate values.

Bas Spitters (Catholic University of Nijmegen, The Netherlands)
A constructive converse of the mean value theorem

Let $f$ be a continuously differentiable function on $[a,b]$. If $c$ in $(a,b)$, then there are $\alpha$ and $\beta$, such that $(f(\beta)-f(\alpha))/(\beta-\alpha)=f'(c)$.}
Assuming some weak conditions Tong and Braza~[Tong, J.\ and Braza, P., A converse of the mean value theorem., Amer. Math. Monthly, December 1997, pp 939--942] proved this theorem using classical reasoning. We will give a constructive proof. [This is joint work with Wim Veldman.]

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Yeneng Sun (National University of Singapore, Singapore)
Asymptotic, Hyperfinite and Continuum Models

The transfer principle guarantees a direct relationship between hyperfinite and asymptotic large finite models. Hyperfinite models can always capture the asymptotic nature of some large finite phenomena, which cannot be analyzed in the usual continuum setting. Several examples in probability theory and game theory are given to show how the hyperfinite framework works well, and why the Lebesgue continuum model fails. These include a systematic study of stochastic independence from a new measure-theoretic point view and some new game-theoretic results in terms of the existence of pure-strategy equilibria. It is also pointed out that a hyperfinite Loeb measure can be seen as some measure on the unit interval. So it goes back to the usual continuum again, but only with different measure structures.

Rudolf Taschner (Technische Universität Wien, Austria)
Real numbers and functions, exhibited in dialogs

It was the late Paul Lorenzen who rediscovered the ancient Socratic method of arguing and proving by dialogs, and who applied it to definitions and theorems of mathematics. In doing so, the differences between the fundamental assumptions of classical mathematics, of recursive constructivism, of Bishop's constructive mathematics, and of intuitionistic mathematics become apparent, especially in connection with various definitions of real numbers, of sequences, of functions. Even attempts to introduce infinitesimals seem to fit into this dialogical frame.

Sebastiaan Terwijn (Universität München, Germany)
Effective Measure Theory

We discuss several aspects of effective measure theory, especially in connection with sets in $Delta^0_2$. We use Lutz's framework of resource bounded measure theory, where sets of measure zero are defined by effectively computable martingales. We discuss measures resulting from various computability restraints on the martingales.

Wim Veldman (Catholic University of Nijmegen, The Netherlands)
On some sets that are not positively Borel

In intuitionistic analysis, a subset $A$ of a Polish space $X$ is called positively Borel if and only if $A$ may be obtained from basic open subsets of $X$ by means of the operations countable union and countable intersection. One is not allowed, during the construction of a positively Borel subset of $X$, to use the operation of taking the complement of a given set. We consider three subsets of Cantor space ${\cal C} = \{0,1\}^{\Bbb N}$ and show that they are not positively Borel.
The first one is the set Bar, consisting of all $\alpha$ in ${\cal C}$ such that for all $\gamma$ in Baire space ${\cal N} = {\Bbb N}^{\Bbb N}$ there exists $n$ such that $\alpha (\overline\gamma n) \ne 0$. ($\overline \gamma n$ is the natural number coding the finite sequence $\langle \gamma (0) , \ldots , \gamma (n-1)\rangle$). An element of Bar may be said to code a bar in ${\cal N}$.
The second one is the set König consisting of all $\alpha$ in ${\cal C}$ such that there exists $\gamma$ in ${\cal C}$ satisfying, for all $n$, $\gamma(n) \le \gamma(n+1)$ and $\alpha (\overline\gamma n) = 0$. An element of König may be said to make true the conclusion of a rather weak form of König's Lemma.
The third one is the set Almostfinite consisting of all $\alpha$ in ${\cal C}$ such that for every strictly increasing $\gamma$ in ${\cal N}$ there exists $n$ such that $\alpha(\gamma (n)) = 0$. An element of Almostfinite is very close to being the characteristic function of a finite subset of ${\Bbb N}$.
Observe that from a non-intuitionistic point of view Bar is $\boldsymbol{\Pi}_1^1$-complete, but König is closed, and Almostfinite is a countable union of closed sets.

Hans Vernaeve (University of Gent, Belgium)
Reducing distributions to hyperreal functions

It is investigated how distributions, singular objects as they are, can be represented by hyperreal $C^\infty$-functions, acting as the kernel of a regular distribution. For that purpose, methods from constructive approximation are used.

Frank A. Waaldijk (Catholic University of Nijmegen, The Netherlands)
The hidden presence of the fan theorem in the constructive definition of `continuous function on the continuum'

In 1967 Bishop [Bishop67] introduced what we will call BISH: a variety of constructive mathematics which adopts much of Brouwer's intuitionistic views, but at the same time rejects certain deep insights such as the fan theorem FT and the axiom of continuous choice \acoo.
Bishop maintained that in the context of analysis one could do without FT and axioms of continuity by restricting one's attention to continuous functions known to be uniformly continuous on compact subspaces of their domain. Only these functions deserved his predicate `continuous', other functions if existent did not merit much attention. It therefore seems that intuitively at least Bishop agreed with Brouwer's analysis which led Brouwer to put forward FT. One can show that this axiom does nothing more than bring about the same restriction on continuous functions (this talk will focus on such a proof). The following question then perhaps merits consideration: if one intuitively agrees with all the consequences of an axiom, then why not simply accept the axiom?

In this talk we show that the existence of any definition of `continuous function' satisfying the above property `uniformly continuous on compact subspaces' and a rather minimal list of other conditions is equivalent within BISH to the fan theorem FT. This throws a fundamental light on both BISH and FT. The main theorem in this respect is:

Theorem Within BISH the following statements are equivalent:

1. The fan theorem FT.

2. There exists a class of real-valued functions called `kontinuous' functions such that:
(a) If f is a uniformly continuous real-valued function defined on \zort, then f is kontinuous.
(b) If f and g are kontinuous functions such that Ran(f)\subseteq Dom(g), then the composition gof is kontinuous.
(c) If f is a kontinuous function defined on [0,1], then f is uniformly continuous.
(d) The function $x\longmapsto\frac{1}{x}$, defined on R^+, is kontinuous.

We obtain a fundamental corollary concerning the two different definitions of `continuous function' which are currently being used in BISH, namely the definition in [Bishop1967] (repeated in [Bishop\&Bridges1985]) and the definition in [Bridges1979]. To formulate this corollary let us call the first definition `continuous^{BIS}' and the second definition `continuous^{BRI}'.

Corollary Within BISH the following three statements are equivalent:

1. The fan theorem FT.

2. If f is a continuous^{BIS} function from [0,1] to R^+, then the composition $\,x\longmapsto\frac{1}{f(x)}\,$ of f with the continuous^{BIS} function $x\longmapsto\frac{1}{x}$ is continuous^{BIS}.

The function $x\longmapsto\frac{1}{x}$, defined on (R^+,d_R), is continuous^{BRI}.

Manfred Wolff (Universität Tübingen, Germany)
Discrete approximation of unbounded operators and the approximation of their spectra

Uniform and strong approximation of an operator $A$ by a sequence of bounded operators $(A_n)$ is a special case of discrete approximation used in numerical analysis. We introduce this notion which in fact is a generalization of a corresponding notion treated by Stummel, Vainikko, Reinhardt, and others. The great progress in this field was possible by the introduction of the notion of the $\varepsilon-$ psesudospectrum $\sigma_\varepsilon(B)$ of an operator $B$ due to L. Trefethen. It is the complement of the $\varepsilon$--resolvent set $\rho_\varepsilon(B) = \{z \in \rho(B):\|(z - B)^{-1}\| < 1/\varepsilon \}$ for given $\varepsilon > 0$. The main theorem says, that if $(A_n)$ converges discretely to the closed unbounded operator $A$ then for every compact set $K$ of the complex plane the distance $d(\sigma_a(A) \cap K, \sigma_\varepsilon(A_n)$ converges to $0$. Here $\sigma_a(A)$ denotes the approximative point spectrum of $A$. Refinements of this theorem as well as applications to pseudo--differential operators are also given. The proofs seem up to now only possible in a reasonable way by nonstandard analysis.

Jiang-Lun Wu (Universität Bochum, Germany)
On hyperfinite integral representation of Euclidean random field measures

Let $F$ be a generalized white noise with infinite divisible probability law on ${\cal D}'(R^d)$}. In recent years, the Euclidean random field $X$ over $R^d$ obtained via


has been successfully used to construct new nontrivial relativistic models in axiomatic quantum field theory (while when $F$ is Gaussian white noise $X$ is a generalized free field for any $\alpha\in(0,1)$ and in particular $X$ is the well-known Nelson's free field if $\alpha={1\over2}$). In this talk, a hyperfinite lattice construction of $X$ will be presented. This provides, via Loeb measure structure, a path integral realization of the probability measure of $X$ and thus gives an interaction picture of the new field models. This is a joint work with Sergio Albeverio.

Julia Zink (Pontificia Università Gregoriana, Città del Vaticano)
Peirce and the continuum from a philosophical point of view

Peirce, an American philosopher and mathematician who lived at the turn of this century, used the concept of continuity for the philosophical description of perception, reaction, and representation. His continuum is of greater `cardinality' than any set, and contains infinitesimals.

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