I'm sympathetic to the Aristotelian view that potential infinity makes sense while actual (completed) infinity doesn't. However, I also find transfinite set theory to be fascinating, and I'm under the impression that this domain of mathematics is founded upon the idea of performing calculations with completed infinities. So for a while I felt like I had to "pick sides", so to speak.
Upon reflection, however, this doesn't really make sense. Formalists and fictionalists ignore or deny the reality of mathematical objects without ignoring/denying mathematics as an activity, so why should my ontological reservations matter when it comes to working with "higher infinities"? Clearly there is something interesting and systematic going on in this domain.
It seems like my difficulty with transfinite set theory is merely semantic: I don't think it makes sense to talk about adding one to $\omega$ if the symbol refers to a completed infinity, but there's nothing wrong with adding one to $\omega$ as long as $\omega$ means something else. My question here is basically: what are my options for that "something else"?
(As an example: one thing I'm wondering is if it's possible to understand transfinite mathematics as operating on potential infinities instead of actual infinities.${}^1$ Perhaps we could understand $\omega$ not as a completed object but as an endless process.${}^2$ Is there a "process" interpretation of transfinite set theory?)
Edit: What about coalgebras? Is there a coalgebraic version of transfinite set theory that treats all completed infinities as incomplete processes?
${}^1$ Potentially relevant: Can we formally distinguish between actual and potential infinities?
${}^2$ What would it mean to "add" and "multiply" processes? Can we make sense of the countable/uncountable distinction in a process context?
The hyperreal numbers do what I believe you are trying to do. While there are many different notations, I prefer the one that uses $\epsilon$ for a standard marker for an infinitesimal, and $\omega$ as a standard marker for an infinity, such that $\epsilon = \frac{1}{\omega}$.
This makes infinities that act semantically basically like uncompleted infinities. You can do arithmetic with them in the ordinary sense, just treating $\omega$ and $\epsilon$ as if they were variables, but then using the "standard part" function (std(x)) to reduce it to a real if you need.
In this system, $2\omega + 1$ is a different value than $\omega - 1$. They can also be put into ratio with one another and cancel: $\frac{5\omega}{10\omega} = \frac{1}{2}$.
You can even do things like $\frac{\omega^2 + 2}{\omega^2}$ which yields $1 + \frac{2}{\omega^2} = 1 + 2\epsilon^2$, whose standard part is just $1$.
This also allows you to assign values to certain divergent series, essentially treating them as if they were convergent to a hyperreal value. For instance, $\frac{1 + 3 + 5 + \ldots}{1 + 2 + 3 + \ldots} \simeq 2$.