Paradoxes in a Semantic Perspective
Let us take Russell’s set of all sets that do not contain themselves as a paradigm of paradox. We assume that there is such a set, and if so, it has to be either a member of itself or not. But, as we know, either way leads to a contradiction. However, this is not the paradox. So far, the argument is just like a reductio ad absurdum proof, and the conclusion to draw is that our assumption was wrong, i.e. there is no such set. The paradox enters when we nevertheless insist that there has to be one — our intuitions about the existence of sets are so strong that we are not prepared to give them up even if they lead to a contradiction. And we cannot deny that they do — the logic of that part is impeccable.
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