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Given any table of all the natural numbers, one can construct a number not in the table, simply by chosing something other then the n'th character in each number given, (ofcause chosing anything when no character apears).

Can someone clearify this for me? So far my conclusion is that either my textbooks are not being rigid enough in their proofs or the only thing cantors diagonal proof realy proves is that it's absurd to talk about a complete list of even a countable set.