Theory about infinity

Question

Is there any theory that takes account of the difference in, for example, $1 \over 0$ and $-\ln 0$ and treat them as different infinities with different notations (and maybe useful theorems)? Cantor's theory only specializes in set cardinality but is there a more general theory on the concept of infinity itself?

Answer 1

Your comment suggests that what you are looking for is the theory of growth rates of functions.

See, for example, the Hardy hierarchy, although you may be interested in fast-growing hierarchies in general.

In the area of the abstract study of growth rates, you may be interested in Hausdorff gaps, or in some of the related cardinal characteristics of the continuum (e.g. the dominating and bounding numbers).