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.
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.