Consider an arbitrary probability distribution P, and the smallest integer (or the lexicographically least object) x such that P(x) < 1/3^^^3 (in Knuth’s up-arrow notation). Since x has a short description, a universal distribution shouldn’t assign it such a low probability, but P does, so P can’t be a universal distribution.
The description of x has to include the description of P, and that has to be computable if a universal distribution is going to assign positive probability to x.
If P has a short computable description, then yes, you can conclude that P is not a universal distribution. Universal distributions are not computable.
If the shortest computable description of P is long, then you can’t conclude from this argument that P is not a universal distribution, but I suspect that it still can’t be a universal distribution, since P is computable.
If there is no computable description of P, then we don’t know that there is a computable description of x, so you have no contradiction to start with.
The description of x has to include the description of P, and that has to be computable if a universal distribution is going to assign positive probability to x.
If P has a short computable description, then yes, you can conclude that P is not a universal distribution. Universal distributions are not computable.
If the shortest computable description of P is long, then you can’t conclude from this argument that P is not a universal distribution, but I suspect that it still can’t be a universal distribution, since P is computable.
If there is no computable description of P, then we don’t know that there is a computable description of x, so you have no contradiction to start with.