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.

>>7
I don't know if I understand.  In the third line, what's w or "f(i)(x)".  F is going to have higher cardinality than the reals, so if the "c" in the first line is the complex numbers, there is no such bijection.

>>8
>>9
You can't do the coin flip thing, since you're not guaranteed to ever get a finite number.  At some point, you'd have to have chosen a finite number of digits, and get nothing but zeros from then on.



I'll post the answer later today.  If you want a hint, look at this: http://en.wikipedia.org/wiki/Vitali_set