Obviously paradoxes don't have to be actual contradictions, merely results which fail to conform to our intuitions (e.g. the Banach-Tarski paradox), but in my e

Obviously paradoxes don't have to be actual contradictions, merely results which fail to conform to our intuitions (e.g. the Banach-Tarski paradox), but in my experience when the term is used in the context of the foundations of mathematics, it does mean that a contradiction is derivable within a foundational theory (the class paradoxes being the obvious case in point). This is not a complaint, I'm just explaining why I took your remark in slightly the wrong way. Working mathematicians ignore the shortcomings of ZFC because foundational issues just aren't what they concern themselves with day-to-day; I'm sure you're aware of the remark that mathematicians are platonists during the week and formalists at the weekend. It's generally left to logicians and philosophers of mathematics (two tribes which, while not coextensional, have a rather large intersection) to worry about these things. Certainly according to structuralists like Shapiro, this is not actually a problem. [1] [1] Shapiro 1997, http://bit.ly/3om0CO http://bit.ly/3om0CO