Usually a paradox comes up when the system deductively lets certain cases derive contradictory propositions. One famous paradox (although not a set theory paradox) is called "The Liar Paradox": when a liar says "I’m always lying", is he lying or not? If he is lying then that means that he tells the truth (not always lying), but if he is telling the truth, then he is lying. It is a paradox so common since antiquity, that this appears in the Bible (Titus 1:12).
<a href="http://pmrb.net/blog/2011/04/20/a-journey-to-platonism-with-edmund-husserl-8/">A Journey to Platonism with Edmund Husserl — 8</a>
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