What is the difference between a counter-intuitive statement and a paradox? In mathematics and logic, what is the difference between a counter-intuitive statement and a paradox? For example, what differs something like the Banach-Tarski theorem or Gabriel's horn from something like Russell's paradox?

A counter-intuitive statement is either true or false like a normal statement, but surprising in the sense that one would expect a different outcome.

A paradox is undefined, in the sense that if one accepts it, it would be in contradiction to some other statement (an antinomy). It gives the dilemma to drop either the paradox or those statements which are in contradiction to the paradox (in that case it would be either true or false).