You should be able to solve this.

Create a method to choose a random integer between 0 and infinity such that no integer is more likely to be chosen than any other.

Yes it is possible, I don't care what your probability book says (Notice I didn't even use the word "probability").  You may assume the axiom of choice.

>>8
>You won't ever finish

The axiom of choice says you can ignore that little detail; it just says it is possible to get infinitely many coin flips, or flipping an infinite number of coins. (You can see now why the axiom is controversial.) Other than that you have the solution.

>>1
Isn't this basically a straightforward application of the axiom of choice? You have an infinite set of bins, each containing the digits 0-9. Select one from each bin. Concatenated together, the digits form a number. More formally, value = sum{n=0}{inf}(b_n * 10^n) where b_n is the digit selected out of the nth bin. This is the same solution as >>8 with a different radix.

This is a possible algorithm, though I realized I haven't really proven that the probability is equal. My set theory is a little rusty.