# Structure of infinities and infinitesimals

It is probably obvious that my education in math is very small. However, it is one of the subjects I enjoy reading. So, if this is too far off the wall, I ask your forgiveness.

I understand infinity as the name of a process. I see it as if there is an x, then there is an x+1 and this is possible for any x. So, infinity is not a number. It does not identify "how many". Correct or not, this allows me some understanding.

Infinitesimals, on the other hand, I had always thought of as very small real numbers. I have read a discussion about infinitesimals that seem to indicate that, like infinity, infinitesimals are a process in the way I think of infinity as a process. Infinitesimal does not identify "how many" or "how much". Similar to infinity, there is a statement that for any x, there is an x - y such that (x - y) < x.

1. Infinity, even countable infinities, do not have a limit. That is, there is no y = x + 1 such that that y + 1 does not exist.

2. Infinitesimals have similar structure in that for any x there exists y such that 0 < y < x. There is no limit.

The two explanations are not intended to be in any way solid math. They are only descriptions that lead to my question. There are different "sizes" of infinities, R and N for example.

Is there a similar structure for infinitesimals?

I guess that you refer to the cardinalities of the sets $$\mathbb N$$ and $$\mathbb R$$, which are actually little comparable to what we denote by $$\infty$$, except that they are all intuitively "infinite".