Why do we use 2 different types of infinity to define the same infinitesimals? I read in the book "Calculus with infinitesimals" (Efrain Soto Apolinar) that $dx=1/N$ and $N$ is the number of elements of the set of the natural numbers (letter $N$ is used to indicate the cardinality of the set of natural numbers).
In other source I read that for hyperreal numbers "$\varepsilon = 1/\omega$" and $\omega$ is number greater than any real number.
Why do we use 2 different types of infinity to define the same infinitesimals?
Thanks.
 A: It's a good question, I think. There is a formulation of this, which skirts around the issue, called Nonstandard analysis, and instead concentrates on the properties of infinitesimals, rather than what they are.
However, I think your question is more about why it doesn't matter which kind of infinity you use, when you work with functions that converge to $0$ - perhaps an intuitive way to look at it would be to say that because $\aleph_0$, the countable infinity, is smaller than $\aleph_1$, the uncountable one, you might argue that $\frac{1}{\aleph_0} \ge \frac{1}{\aleph_1}$; but $\frac{1}{\aleph_0}$ is already $0$, and we don't have another zero that is somehow smaller, so there isn't any difference between the outcome of using one kind of infinity rather than another.
The really important thing about infinitesimals, IMO, is the observation that the value of $\frac{0}{0}$ depends entirely on how you arrive at the $0$ on top and the one at the bottom - that is, how you converge to $0$.
Now, I'm sure there are many people with a more firm grasp of things, who will have felt their toes curl up on reading my explanation, but I don't think is entirely wrong. Sometimes a good lie is better than a bad truth ;-)
