I find it funny, that it works even in continuous case: suppose that we have probability density defined in R^n (or any other set). Then whatever bijection F:R <--> R^n we apply, the integral of probability density on that path should converge, therefore p(F(x)) goes to zero faster than 1/x. :)
Also, look: suppose the “real” universe is a random point x from some infinite set X. Let’s say we are considering finite set of hypotheses “H”. Probability that random hypothesis h € H is closest to x is 1/|H|. So the larger H is, the less likely it is that any particular point from it is the best description of our universe! Which gives us Occam’s razor in terms of accuracy, instead of correctness, and works for uncountable sets of universes.
And in this case it is almost surely impossible to describe universe in a finite amount of symbols.
I find it funny, that it works even in continuous case: suppose that we have probability density defined in R^n (or any other set). Then whatever bijection F:R <--> R^n we apply, the integral of probability density on that path should converge, therefore p(F(x)) goes to zero faster than 1/x. :)
Also, look: suppose the “real” universe is a random point x from some infinite set X. Let’s say we are considering finite set of hypotheses “H”. Probability that random hypothesis h € H is closest to x is 1/|H|. So the larger H is, the less likely it is that any particular point from it is the best description of our universe! Which gives us Occam’s razor in terms of accuracy, instead of correctness, and works for uncountable sets of universes.
And in this case it is almost surely impossible to describe universe in a finite amount of symbols.