Fitch’s paradox of knowability is one of the fundamental puzzles of epistemic logic. It provides a challenge to the knowability thesis, which states that every truth is, in principle, knowable. The paradox is that this assumption implies the omniscience principle, which asserts that every truth is known. Essentially, Fitch’s paradox asserts that the existence of an unknown truth is unknowable. So if all truths were knowable, it would follow that all truths are in fact known. The paradox is of concern for verificationist or anti-realist accounts of truth, for which the knowability thesis is very plausible, but the omniscience principle is very implausible.
A formal proof of the paradox is as follows.
Suppose p is a sentence which is an unknown truth; that is, the sentence p is true, but it is not known that p is true. In such a case, the sentence “the sentence p is an unknown truth” is true; and, if all truths are knowable, it should be possible to know that “p is an unknown truth”. But this isn’t possible, because as soon as we know “p is an unknown truth”, we know that p is true, rendering p no longer an unknown truth, so the statement “p is an unknown truth” becomes a falsity. Hence, the statement “p is an unknown truth” cannot be both known and true at the same time. Therefore, if all truths are knowable, the set of “all truths” must not include any of the form “something is an unknown truth”; thus there must be no unknown truths, and thus all truths must be known.
The proof has been used to argue against versions of anti-realism committed to the thesis that all truths are knowable. For clearly there are unknown truths; individually and collectively we are non-omniscient. So, by the main result, it is false that all truths are knowable. The result has also been used to draw more general lessons about the limits of human knowledge. Still others have taken the proof to be fallacious, since it collapses an apparently moderate brand of anti-realism into an obviously implausible and naive idealism.
The Logical Place 
I’m not sure what this adds to Goedel’s theorem. Consider the possibility that Goldbach’s conjecture, “Every even number is the sum of two primes”, is one of the statements that can be neither proved nor disproved; (Goedel of course proved that within any mathematical system rich enough to include arithmetic, such statements must exist.) In that case, it would be true that the truth of Goldbach’s conjecture was unknowable. But if we knew that the truth of Goldbach’s conjecture was unknowable, we would know that it could not be refuted, meaning that it was impossible to produce a counterexample, meaning that we would know it to be true.
Moreover, I cannot for the life of me see the problem in asserting that there are some truths that are unknowable.
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