I've always found dedekinds construction (where sqrt 2 is represented by the set of rationals {x in Q : x^2 < 2}) to be a nice natural place where powersets of

I've always found dedekinds construction (where sqrt 2 is represented by the set of rationals {x in Q : x^2 < 2}) to be a nice natural place where powersets of countable infinite sets come up (in addition to being a great way to construct R)