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@Alex ah, my mistake.  Sarcasm is difficult to pick up on in email.

@Dan technically a slightly higher standard than "can be of infinite
length" is required. The *rationals* can be of infinite length but all of
them terminate or repeat in a periodic fashion and therefore are still
countable.  If they are infinite without repetition (as in the irrational
or as you mentioned, the reals) then they are uncountable.