If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should be posted in Discussion, and not Main.
4. Open Threads should start on Monday, and end on Sunday.
Sure. It's called super-tasks.
https://en.wikipedia.org/wiki/Supertask
"a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time."
You can't avoid supertasks, when you endorse infinity.
Therefore, I don't.
What you are doing in many ways amounts to the 18th and early 19th century arguments over whether 1-1+1-1+1-1... converged and if so to what. First formalize what you mean, and then get an answer. And a rough intuition of what should formally work that leads to a problem is not at all the same thing as an inconsistency in either PA or ZFC.