Okay, per Asaf's clarification (thanks!) I think this answer is correct, but I can not immediately revoke my downvote (the system will not let me). anon's comment is a legitimate concern (what about negative rationals? why can we assert the result that $|\mathbb Q| = |\mathbb N|$, which I would normally prove after proving $|\mathbb{N imes N}| = |\mathbb N|$?)