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.

I'm not OP here, but you're all being fucking retard (Including OP)

As far as I can tell here's OP's argument

I take sets A_i = {0,1,2,3,4,5,6,7,8,9} for all natural numbers i.

I then define a real number in [0,1] by picking a_i a representative from each A_i and forming the number

a=0.a_1a_2a_3a_4......

>>53 I think here you misunderstood what he was saying

I've also partitioned [0,1] into a countable number of sets, labelled V_1, V_2..... using the vitali sets

Then I look at which V_i a is in.

This argument is fine, it all works, not certain it works in the way he wants it to, the very point of vitali sets is all ideas of "probability" is useless for them.


Now, one actual criticism here is that it doesn't use the axiom of choice.

Sure we have an infinite number of sets, and we have to choose an element of each, but I can explicity construct a choice function f(i)=0, ta-fucking-da.

Also the axiom of choice, even if it were applicable here, wouldn't make your choices evenly distributed along [0,1] for any reason.

Luckily you can just do this with normal probability, so we're all fine, picking a random number in [0,1] is just define random variables X_i, all unformly distributed on {0,1,2....9}


and define X = X_1/10 + X_2/10^2 + X_3/10^3.....

nothing to do with the axiom of choice, the only place you use it here is in constructing the vitali sets