The problem of induction. No one has answered Hume, except Kant and he dodged the question. This is the big question, underlying integration of QM/GR, the nature of time, P=NP?, AI, a stable credit-money system, scientific ethics, chicken or egg?, etc, etc. Once this is resolved, I feel sorry for future generation (not really) because its "only" applications forever after.
ET Jaynes solved it indirectly in his book Probability Theory: The Logic of Science(1). As a tl;dr, it turns out that knowledge--as it's defined in the problem of induction--doesn't exist.
I agree Jaynes made some significant, important progress. However, the conclusion that the problem does not exist comes from Kant, its a dodge. The problem is still open because Kant's theory of knowledge is a denial of ALL knowledge.
