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It is actually quite rare that a set contains its own size as an element. E.g. {0,1,2,3} also has size 4, but does not contain 4 itself. That {1,2,3,...} does not contain infinity should not be surprising. "..." essentially means that the set is generated by adding 1 to any number already in the set. Is infinity the result of adding 1 to a number in the set? If infinity-1 is in the set. Is infinity-1 in there? If infinity-2 is. ... This gets you a set {1,2,3,...,infinity-2, infinity-1, infinity, infinity+1, ...} But you can do the same with foo and get the set {1,2,3,...,foo-2, foo-1, foo, foo+1, ...}. So if infinity is in {1,2,3,...}, the same should be true for foo or anything else. This obviously doesn't make sense, so {1,2,3,...} is defined to be the smallest possible set containing 1 and n+1 for each n in the set. |