Martin Gardner wrote in his career at Scientific American that (paraphrased): a paradox, being self contradictory, is wrong. There is either an error in the assumption or the resolution.

If the assumption is false or illogical, the rest of the paradox is pure fantasy. I good example is the famous island of truth-tellers and liars who exclusively lie or tell the truth. One man says, "I'm lying." Which type is he? From the given conditions, he can't be either, so the assumptions are wrong. One of these things must be: the man never said that, or there is no such island (bingo!).

If the resolution is faulty, it can be any step after the given initial conditions. Mathematics has several operations called irreversable steps, like taking the square root of a number or dividing by a variable. In the latter, if the variable in question solves out to zero, the proof is invalid.

Paradoxes don't exist in the real world, because things are rarely cut-and-dry into boolean logic.