I must be missing something. Your argument makes no sense at all to me. It is very simple to construct a countable set of hypotheses of various lengths, all of which have prior probability exactly 1/2. The hypotheses denote events in a universe consisting of an infinite sequence of coin flips, for example.
Hypotheses in this sense are exclusive. If you prefer, consider hypotheses to be descriptors of all possible data one will get from some infinite string of bits.
Furthermore, I must be misreading your definition of h(n), because as I read it, h(n) goes to 1 as n goes to infinity. I.e. it becomes overwhelmingly likely that at least one shorter-than-n hypothesis is correct.
Sorry, yes, h(n) should be for statements with length at least n not at most n.
Assume we have a countable set of hypotheses described in some formal way with some prior distribution such that 1) our prior for each hypothesis is non-zero 2) our formal description system has only a finite number of hypotheses of any fixed length. Then, I claim that that under just this set of weak constraints, our hypotheses are under a condition that informally acts a lot like Occam's razor. In particular, let h(n) be the the probability mass assigned to "a hypothesis with description at least exactly n is correct." (ETA: fixed from earlier statement) Then, as n goes to infinity, h(n) goes to zero. So, when one looks in the large-scale complicated hypotheses must have low probability. This suggests that one doesn't need any appeal to computability or anything similar to accept some form of Occam's razor. One only needs that one has a countable hypothesis space, no hypothesis has probability zero or one, and that has a non-stupid way of writing down hypotheses.
A few questions: 1) Am I correct in seeing this as Occam-like or is this just an indication that I'm using too weak a notion of Occam's razor?
2) Is this point novel? I'm not as familiar with the Bayesian literature as other people here so I'm hoping that someone can point out if this point has been made before.
ETA: This was apparently a point made by Unknowns in an earlier thread which I totally forgot but probably read at the time. Thanks also for the other pointers.