For a countable set, a uniform probability distribution is also possible by replacing the axiom of countable additivity with finite additivity. See here. It would mean each element in the countable set has probability 0.
This makes sense from the concept of potential infinity: Take a finite set of size n with uniform probability distribution. As n approaches infinity, the probability of each element approaches 0. Under potential infinity, a countable set is just the infinite limit of a growing finite set, so each element must be assigned zero probability. This means it almost surely doesn’t happen, not that it is impossible.
The standard example is an infinite lottery. Insofar such a lottery seems possible in principle, a uniform probability distribution on countable sets must be admitted.
The video linked above also discusses other approaches. The topic has applications in cosmology.
For a countable set, a uniform probability distribution is also possible by replacing the axiom of countable additivity with finite additivity. See here. It would mean each element in the countable set has probability 0.
This makes sense from the concept of potential infinity: Take a finite set of size n with uniform probability distribution. As n approaches infinity, the probability of each element approaches 0. Under potential infinity, a countable set is just the infinite limit of a growing finite set, so each element must be assigned zero probability. This means it almost surely doesn’t happen, not that it is impossible.
The standard example is an infinite lottery. Insofar such a lottery seems possible in principle, a uniform probability distribution on countable sets must be admitted.
The video linked above also discusses other approaches. The topic has applications in cosmology.