Por Imme van den Berg (CIMA, University of Évora).

In the book Radically Elementary Probability Theory [3] Nelson presented an elementary nonstandard axiomatic system, sometimes called ENA (Elementary Nonstandard Axiomatics without external sets). This axiomatic system resembles Peano Arithmetic and assigns the predicate "standard" only to natural numbers. Avoiding measure theory, Nelson developed an alternative approach to advanced stochastics, culminating in a Martingale Central Limit Theorem. Though he claims that it reects versions of all essential variants of the Central Limit Theorem, it has been criticized for not including the Gaussian density function. This particular function uses real constants like e and _, which ENA fails to de_ne. Recently [1] I proposed the system ENE (Elementary Nonstandard axiomatics with External sets), in order to be able to deal with standard real numbers, while remaining reasonably elementary. The real numbers will be generated by Nelson's _axiom - a form of countable saturation he seldom uses in [3] -, and a fragment of Kanovei and Reeken's system HST [2], avoiding the axiom-schemes of the latter. We thus obtain a real closed _eld which enables an interplay between nonstandard analysis and classical analysis for de_nable objects, in particular the Gaussian density function.

References:

• [1] I. van den Berg, Elementary nonstandard axiomatic systems and standard real numbers, 25p., accepted for Advances in Mathematical Modeling in Science, Engineering and Social Sciences, Trends in Mathematics, Springer-Birkhauser.
• [2] V. Kanovei, M. Reeken, Nonstandard Analysis, Axiomatically, Springer (2004).
• [3] E. Nelson, Radically elementary probability theory, Princeton University Press (1987).

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