As long as you can clearly define your starting axioms you can mathematically prove or disprove anything. The physical example would be to just say, suppose a universe with the same properties of ours exist but where the gravitational constant is twice as large prove whether X can occur in that universe.
As far as how randomness is defined I believe that might be field dependent but this is me getting out of my depth (I've only taken a few courses on combinatorics).
Randomness is usually defined in terms of a probability distribution, which is just a measure where the total mass assigned to all elements adds up to 1.
As far as how randomness is defined I believe that might be field dependent but this is me getting out of my depth (I've only taken a few courses on combinatorics).