Ceiling function of an infinitesimal

Question

I was working with infinitesimals and I came across the problem: what is the ceiling function of an infinitesimal? Wolfram Alpha says an infinitesimal equals zero, so therefore the ceiling function of it should equal zero. But what I've seen is that an infinitesimal is not equal to zero but less than any positive real value, so I thought it would be one. What is the right answer?

Answer 1

If you’re working in the hyperreals, the question is meaningful, and the answer follows immediately from the definitions:

• if $\alpha$ is a positive infinitesimal, then $0<\alpha<1$, so $\lceil\alpha\rceil=1$, and
• if $\alpha$ is a negative infinitesimal, then $-1<\alpha<0$, so $\lceil\alpha\rceil=0$.

Answer 2

To respond to a closely related issue raised in a comment following the question, note that applying the floor function to hyperreals plays an important role when one wishes to define the hyperintegers. Namely, a hyperinteger can be defined as a hyperreal number $H$ satisfying $H=\lfloor H\rfloor$. Hyperintegers play a very basic role in the theory. For example, the convergence of a sequence $(u_n)$ to $L$ can be characterized as follows: $u_H\approx L$ for all infinite positive hyperintegers $H$, where $\approx$ is the relation of infinite proximity (i.e., the difference of the two sides is infinitesimal).