Thank you for the feedback. In this post I did equate ZFC with "math" and you're right in saying that set theory is only one possible basis for math.

I think this causes some confusion if you know first order logic and set theory. I just meant that this independence of AoC from ZF implies it has no effect on the question whether math (ZF) is consistent. Any contradiction that might result from ZFC is a contradiction in ZF. To me that means it's safe to add AoC to ZF, but of course it's as safe as adding its negation.

All in all, I think Godel's second incompleteness theorem doesn't contradict my statement. Any inconsistent system can prove anything, including its own contradiction. If that happens, AoC is absolved from any crime.

Honestly, as you've noticed, this article is written in pretty simple terms. I'd prefer to go deeper into things but this is really not a platform for such endeavours.

If you want more formal math, literally check any other article. One includes Godel's incompleteness theorems:

https://nseverkar.medium.com/the-diagonalization-argument-53dca529570e