I think you misunderstand what "paradox" means. While it can mean "self-contradictory" it can also mean "contrary to one's expectation." Math uses both, but in very different contexts.

The contradiction is used in proof formulation, specifically to invalidate some claim. I don't think this is what you're implying.

The latter is what it contextually sounds like you're stating; things like the Banach-Tarksi Paradox. There's no self-contradiction in that, but it is an unexpected result and points to the need to refine certain things like the ZFC set theory.

I'd also stress that there are true statements which cannot be proven through axiomatic systems. The Halting Problem is an example of what Godel proved. But that's not contradictory, even if unexpected or frustrating.