Almost all hypotheses have high complexity. Therefore most high-complexity hypotheses must have low probability.

(To put it differently: let p(n) be the total probability of all hypotheses with complexity n, where I assume we’ve defined complexity in some way that makes it always a positive integer. Then the sum of the p(n) converges, which implies that the p(n) tend to 0. So for large n the total probability of all hypotheses of complexity n must be small, never mind the probability of any particular one.)

Note: all this tells you only about what happens in the limit. It’s all consistent with there being some particular high-complexity hypotheses with high probability.

But why should the probability for lower-complexity hypotheses be any lower?

It shouldn’t, it should be higher.

If you just meant ”… be any higher?” then the answer is that if the probabilities of the higher-complexity hypotheses tend to zero, then for any particular low-complexity hypothesis H all but finitely many of the higher-complexity hypotheses have lower probability. (That’s just part of what “tending to zero” means.)

Almost all hypotheses have high complexity. Therefore most high-complexity hypotheses must have low probability.

(To put it differently: let p(n) be the total probability of all hypotheses with complexity n, where I assume we’ve defined complexity in some way that makes it always a positive integer. Then the sum of the p(n) converges, which implies that the p(n) tend to 0. So for large n the total probability of

allhypotheses of complexity n must be small, never mind the probability of any particular one.)Note: all this tells you only about what happens in the limit. It’s all consistent with there being some particular high-complexity hypotheses with high probability.

But why should the probability for lower-complexity hypotheses be any lower?

It shouldn’t, it should be higher.

If you just meant ”… be any higher?” then the answer is that if the probabilities of the higher-complexity hypotheses tend to zero, then for any particular low-complexity hypothesis H all but finitely many of the higher-complexity hypotheses have lower probability. (That’s just part of what “tending to zero” means.)