Huh? A subset of an uncountable can be countable, obviously.
Integers are a countable subset of reals.
The set of function X -> Y is size of Y to the power size of X. A small set of functions on X is the set of boolean functions on X is equivalent P(X), as it is isomorphic to the set of subset of X (each function f_i is just the function "is x a member of X_i?")
The reason that P(integers) is uncountable is due to a diagonalization argument, given any countable list of sets of integers, it is easy to construct an set of integers not in that list (just take the union of each sets smallest (positive) non-member, which is possible since strict ordering is always a well-defined concept on a countable set)
Hah, yeah, I wasn't paying attention there. I realized my argument was flawed in the shower and was faced with an unpleasant dilemma--do I run, dripping, back to my room to fix it, or do I finish my shower? I naturally took the second choice :P.
The set of function X -> Y is size of Y to the power size of X. A small set of functions on X is the set of boolean functions on X is equivalent P(X), as it is isomorphic to the set of subset of X (each function f_i is just the function "is x a member of X_i?")
The reason that P(integers) is uncountable is due to a diagonalization argument, given any countable list of sets of integers, it is easy to construct an set of integers not in that list (just take the union of each sets smallest (positive) non-member, which is possible since strict ordering is always a well-defined concept on a countable set)