Paradoxes arise in mathematics; a famous one being Russells; usually its taken as a sign that the theoretical ediface needs to change. The difficult question is how; for example Russell developed his theory of types to dissolve the threatening nature of his.
But can there be actual paradoxes? For example, surely we can take it for granted that the individual & collective intelligence of humans is limited; thus there is a limit to ingenuity in resolving difficult questions; taking these questions to the 'limit' provides a sense of questions whose nature is beyond the ability of humans, alone or enhanced (ie by computers) to understand; they will then be to us refractory and paradoxical; possibly in every possible way.
If they are not actual paradoxes; then what are they? One might say that it is sufficient to posit that such a question is paradoxical; one must prove that it cannot be made 'unparadoxical'; but can this be possible when we have placed it beyond any human endeavour to decipher? Under such circumstances the question of proof itself, appears to becomes irrelevant.