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I'm taking a class that's covering cardinalities, and I was introduced to Cantor's diagonal argument today, and I'm having trouble following the logic. The theorem states, "If s, s, … , s[n], … is any enumeration of elements from T, then there is always an element s of T which corresponds to no s[n] in the enumeration." This sounds more like a logical paradox than a theorem to me.
Specifically, it states that "By construction, s differs from each s[n], since their nth digits differ. Hence, s cannot occur in the enumeration." (from Wikipedia)
However, if s[..] is "any enumeration of elements" in T, then it follows that any s created by choosing elements of T, by whatever method, must occur as some s[n], because s[..] represents every combination of digits possible, right?
This is where I can no longer follow the logic of the proof. Where does the assumption come from, that this diagonal sequence of digits is somehow special and doesn't occur anywhere else in s[..]? Wouldn't that invalidate the first statement of the theorem?
Sorry if this question seems obvious or stupid, but I can't find an explanation that doesn't seem (to me) to invalidate itself.