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Originally Posted by Yakk
Actually, we can't.
See Gödel's incompleteness theorem. We can't describe the integers completely, consistantly and decideably. The integers cannot be described.
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I think you may be talking about number theory. I'm not. I'm simply saying that the integers can be described by a set in such a way as there is no confusion about them. We can describe an object such as the set of integers without needing to describe every possible relation of the integers to each other. The former is describing the integers. The latter is number theory.