How to prove that there are nilsquare infinitesimals that indistinguishable from zero?

Question

As is known, non-zero infinitesimals exist. It can be proved.

In the book "A Primer of Infinitesimal Analysis" John Bell introduced infinitesimals that indistinguishable from zero. He did it implicitly.

How did he know that such infinitesimals exist? He didn't prove it.

How to prove it?

Thanks.

Answer 1

This is a non-trivial matter. There is a sketch in the Appendix to Bell's book

Bell, John L. A primer of infinitesimal analysis. Second edition. Cambridge University Press, Cambridge, 2008. xii+124 pp. ISBN: 978-0-521-88718-2; 0-521-88718-6

Here on page one finds:

In this appendix we sketch the construction of models for smooth infinitesimal analysis. We assume here an acquaintance with the basic concepts of category theory (see Mac Lane and Moerdijk, 1992 or McLarty, 1992). The central concept in the construction of such models is that of a topos. To arrive at the concept of a topos, we start with the familiar category Set of sets whose objects are all sets and whose maps are all functions between them.