There are undecidable statements (about arithmetic)... There are true statements lacking proof. There are also false statements about arithmetic the proof of whose falsehood is impossible; not just impossible for you and me but for a computer of any capacity or other forms of rational processing. We'll never have a computer, then, that will work as a mathematically-omniscient device. By that I mean a computer such that every question that has a mathematically-oriented theme having an answer truthfully can be answered by such a device. Calculators demonstrate the concept but are clearly not mathematically-omniscient: you ask the calculator what is 2+2 and press a button and "presto" you get an answer. What I'm talking about would be questions like "is the set of rational numbers equal in size to the set of real numbers", and get the correct answer. So we will never have such a computer no matter what its capacities are, even if computer encompasses the entire human brain. Unfortunately, that means that even for humans, we will never know everything about math. Unless something weird would happen and we suddenly had infinite capacities; that might change the conclusions.
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