Symposion, 16 - 23 May 1999, San Servolo, Venice

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*Petr V. Andreyev*
(Nizhnii Novgorod State University, Russia)

**Definable standardness predicates in Internal Set Theory**

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**

*Nigel J. Cutland* (University of Hull, England)

**
Constructive aspects of nonstandard methods in fluid mechanics
**

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**

$\Phi$ 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
**

*P.V.Andreyev and Evgenii I. Gordon*
(Nizhnii Novgorod State University, Russia)

**Mathematics in Hyperfinite Sets Theory**

[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**

*Chris Impens* (University of Gent, Belgium)

**
Some thoughts on nonstandard geometry
**

*Mohamed Khalouani*
(Université de Franche-Comté, Besançon, France)

**Étude constructive de problemes de topologie pour les reels
irrationnels**

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**

*P. Ekkehard Kopp* (University of Hull, England)

**
Hyperfinite discretisations and convergence in option
pricing 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**

*Xiaoai Lin* (National University of Singapore)

**On the almost independence of correspondences on Loeb space**

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**

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
**

2) If $1_N:

3) $f(\alpha)\in g(\alpha)$ iff $f(\alpha)=h(\alpha)$ for a sequence $h$ with $h(n)\in g(n)$ for all $n\in

4) If $f(n)=\{g(n),h(n)\}$ for all $n\in

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

*Sara Negri* (University of Helsinki, Finland)

**
On the proof theory of constructive ordered fields
**

*Juha Oikkonen* (University of Helsinki, Finland)

**
Some geometric ideas related to Brownian motion
**

*Hervé Perdry*
(Université de Franche-Comté, Besançon, France)

**
Computing in the constructive henselisation of a valued field
**

*Jan von Plato* (University of Helsinki, Finland)

**
The lattice of open intervals of the intuitionistic continuum
**

*Herrmann Render* (Universität Duisburg, Germany)

**Borel measure extensions of measures defined on sub-$\sigma$-algebras**

*Fred Richman* (Florida Atlantic University, Boca Raton)

**Constructive mathematics without choice**

*Frank Rosemeier* (Fernuniversität Hagen, Germany)

**
On Conway-numbers and generalized real numbers
**

*David Ross* (University of Hawaii)

**The constructive content of nonstandard measure existence proofs:
is there any?**

*Monika Seisenberger* (Universität München, Germany)

**Kruskal's tree theorem in a constructive theory of inductive definitions**

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**

*Bas Spitters* (Catholic University of Nijmegen, The Netherlands)

**A constructive converse of the mean value theorem**

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**

*Rudolf Taschner* (Technische Universität Wien, Austria)

**
Real numbers and functions, exhibited in dialogs
**

*Sebastiaan Terwijn* (Universität München, Germany)

**
Effective Measure Theory
**

*Wim Veldman* (Catholic University of Nijmegen, The Netherlands)

**
On some sets that are not positively Borel
**

The first one is the set

The second one is the set

The third one is the set

Observe that from a non-intuitionistic point of view

*Hans Vernaeve* (University of Gent, Belgium)

**
Reducing distributions to hyperreal functions
**

*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'
**

Bishop maintained that in the context of analysis one could do without

In this talk we show that the

1. The fan theorem

2. There exists a class of real-valued functions called `kontinuous' functions such that:

(a) If

(b) If

(c) If

(d) The function $x\longmapsto\frac{1}{x}$, defined on

We obtain a fundamental corollary concerning the two different definitions of `continuous function' which are currently being used in

1. The fan theorem

2. If

The function $x\longmapsto\frac{1}{x}$, defined on (

*Manfred Wolff* (Universität Tübingen, Germany)

**Discrete approximation of unbounded operators and the
approximation of their spectra**

*Jiang-Lun Wu* (Universität Bochum, Germany)

**
On hyperfinite integral representation of Euclidean random field measures
**

$$(-\Delta+m^2)^{\alpha}X=F,\quad\alpha\in(0,1)$$

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**

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