Does "$\varepsilon$ is indistinguishable from zero" mean that $\varepsilon$ is less than any positive real number?

Question

I read book "A primer of infinitesimal analysis" John Bell. Author wrote about infinitesimals which indistinguishable from zero.

Does "$\varepsilon$ indistinguishable from zero" mean that $\varepsilon$ is less than any positive real number?

Thanks.

Answer 1

If $\epsilon$ is a nilsquare infinitesimal, then it will be smaller than any real number $r$ which provably satisfies $r>0$. Indeed, otherwise $\epsilon$ itself will be provably positive! The opposite implication may not necessarily hold. For example, in some models of Synthetic Differential Geometry, one has nilpotent infinitesimals that are not provably nilsquare. For example, one could have provably $\epsilon^3=0$ but $\epsilon^2=0$ is not provable. Such infinitesimals will also be smaller than every strictly positive real number (for the same reason as mentioned above), but as mentioned are not themselves provably nilsquare. As I recall there is an appendix at the end of Bell's book that discusses all sorts of classes of infinitesimals. You could look it up there.