Can numbers smaller than infinitesimals exist?

Question

I have a good idea of infinitesimals to some extent.( A bit of non standard analysis) I am reading the book of keisler on non standard analysis and calculus. I am okay with them all but, if "a" is an hyperreal number then 2a,3a.... Are also infinitesimals right? So can I do,0.5a,0.00009a..... if yes then can much smaller numbers than infinitesimals exist?

Answer 1

If you define $x$ as positive infinitesimal if $x>0$ while $x<y$ for all positive reals, negative infinitesimal if $-x$ is positive infinitesimal, and infinitesimal as either, by definition anything smaller than an infinitesimal is another infinitesimal or $0$, and hence no further concept is needed.

But infinitesimals form a hierarchy if they aren't nilpotent. For example, a positive infinitesimal $\varepsilon$ satisfies $\varepsilon>\varepsilon^2>\varepsilon^3>\cdots>0$, unless $\epsilon^n=0$ for sufficiently large $n\in\Bbb N$.