I recently found Cantor's diagonal argument in Wikipedia, which is a really neat proof that some infinities are bigger than others (mind blown!).
But then I realized this leads to an apparent paradox about Cantor's argument which I can't solve. Basically, Cantor proves that a set of infinite binary sequences is uncountable, right?. But what about a set of all possible binary sequences (infinite or not)? Wouldn't that then be uncountable as well? And if so, doesn't that also imply that the natural numbers are also uncountable?
Note: I looked at other similar questions in this site, which says there is no natural number with an infinite number of digits. I don't understand this explanation, because we need a large number of digits to represent a large number. So why can't we have a interger with infinite digits?