Una via ingenua agli infinitesimi. Trattazione non ortodossa della NSA. [A simple pathto the infinitesimal. An unconventional treatise on NSA]. With a preface by Wim Klein Haneveld and Sileen Ponstein-Troelstra. Translated by Sergio Casiraghi and Lucia Casiraghi.
This book gives a naive (in the sense of [P. R. Halmos, Naive set theory, The University Series in Undergraduate Mathematics, D. Van Nostrand Co., Princeton, NJ, 1960; MR0114756]) introduction to nonstandard analysis. It does not go deep into the logical intricacies of the formalization (this was not the author’s intention), but it does touch on the difficulties that arise in its formalization. For instance, in the final sections of the first chapter, the idea of the ultrapower construction of the reals with infinitesimals is introduced in a very intuitive manner, although the ultraproduct construction itself does not appear. He takes great care to explain the transfer principle and how to use it, the distinction between internal and external sets and the pitfalls of confusing them. He also touches on some philosophical problems related to constructivism versus formalism.