i tend to agree with James. however let's try this construction.
first, I'll use binary, so only 1 and 0.
then I'll construct a transcendental number.
let 0->01 and 1 -> 0
0
01
010
01001
01001010
repeat indefinitely, use this number as our 1.
1 -> 0.01001010..
we know the rational numbers are countably infinite, and doubling the rational numbers is not enough to change countability.
so, i propose we rotate between the list of rational numbers, and xor the list rational numbers with this transcendental number.
prove this doesn't go through every positive number.