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Jul 27, 2011 at 12:00 comment added Carl Mummert @Qiaochu, the same thing works for other logics if we choose the correct semantics for them. Unfortunately the terminology "first-order" has a frightful number of meanings.
Jul 26, 2011 at 20:30 comment added Kaveh @Qiaochu, if we cannot express them in the language that proofs are being written proving them (in the logical/formal sense) is obviously impossible.
Jul 26, 2011 at 20:25 comment added Qiaochu Yuan Well, but it seems to me that (as per the discussion in the comments) the properties the OP is interested in are not first-order.
Jul 26, 2011 at 20:19 history answered Kaveh CC BY-SA 3.0