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Hint 1 : One of these is countably infinite, the other uncountably infinite. Hint 2 : One of these is equivalent to the rational numbers

Hint 1: One of these is countably infinite, the other uncountably infinite.
Hint 2: One of these is equivalent to the rational numbers Q, the other to the reals R.
(5 points) Bonus
The diagonalization argument FAILS in one of these. Why?
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