People most often hear about paradoxes that challenge our notions of truth and falsity, like the Liar Paradox, Curry's Paradox, Russell's Paradox, Berry's Paradox, etc. But just as interesting are the paradoxes that challenges other notions we hold dear, the ones philosophers call "modal" notions: knowledge, possibility, morality. So let me present one of the most famous ones, called Fitch's Paradox of Knowability, and if people find that interesting I can talk about a few other favorites of mine. The question we're dealing with is: Are all true statements knowable? To put it another way, is it possible for there to be some truth which can never be known, no matter how hard you try? Here's an argument that seems to answer this question. Obviously there are some unknown true statements out there; we don't know everything, do we? For instance, either "The Riemann Hypothesis is true" or "The Riemann Hypothesis is false" is one of these statements. In any case, let P be some unknown true statement. Then consider the statement Q, which says "P is an unknown truth." Then Q is obviously a truth. Is it possible for Q to be known? Well, suppose Q were known. Then we would be able to say "I know that Q is true" or equivalently "I know that P is an unknown truth" or in other words "I know that P is true and that P is unknown." But it's impossible for that to be true, isn't it? Because if you knew that P is true, then P would be known, so it would be impossible to know that P is unknown, because P is not unknown, and you can't know a false statement! Thus it's impossible to know Q, so in other words Q is an unknowable truth. So to review, we started with the hypothesis that P is an unknown truth and we got to the conclusion that Q is an unknowable truth. So "there exists an unknown truth" implies "there exists an unknowable truth." Turning this around, "all truths are knowable" implies "all truths are known", which is crazy! Clearly it is possible for there to be some truths which we happen to be unknown right now, but might be discovered in the future. But Fitch's argument above seems to suggest that if you believe that any truth is within our grasp, you have to believe that we already know everything!