I am an inexperienced logician, so I may be completely missing something major in this question. I also may be misconstruing the idea of decidability. However, I was wondering if all 6 of the remaining Millennium Prize Problems are decidable in the sense of Gödel. If any of the associated theories were not decidable, wouldn't that have far-reaching applications in the world of mathematics? Thanks in advance, and I hope that my question makes sense.
closed as off topic by Gerald Edgar, Harry Gindi, Mark Sapir, Zev Chonoles, Mariano Suárez-Alvarez♦ Apr 18 '11 at 17:41
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There are very few results which allow us to know that a mathematical claim will be provable or disprovable within ZFC without actually proving or disproving it. To the best of my knowledge, the only exceptions are theories which have quantifier elimination. Few1 open mathematical problems which people are interested in are of this sort, and none of the Millenium problems are. So any of the Millenium problems could be independent of ZFC (except for the Poincare conjecture, because it has been proved!)
You might be particularly interested in Scott Aaronson's survey on whether or not it is likely that $P \neq NP$ is independent of ZFC.
1 Here is an example of a question which I know is decidable in ZFC, yet whether the answer is "yes" or "no" is open.
This is the a first order question about real numbers, so it is decidable by Tarski's theorem. The analogous result for four dimensional vectors was only obtained in 2003; if you can get the answer for $5$ dimensions, it should be publishable in a good journal. I think this about as interesting a question as one can find which is definitely settleable in ZFC, yet still open. Most questions mathematicians care about are not of this form (and, in my opinion, are much more interesting).
Most of the Millennium Prize Problems are individual problems, with a single yes or no answer. Decidability as a question only really makes sense in the context of a countably infinite family questions, where you can ask whether it's decidable which of those questions should be answered yes.
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