I really appreciate having the examples in parentheses and italicised. It lets me easily skip them when I know what you mean. I wish others would do this.

"Doesn't physics say this universe is going to run out of negentropy before you can do an infinite amount of computation?" Actually, there is a [proposal that could create a computer that runs forever.

It doesn't: for example, the set of strings 0, 10, 110, 1110, 11110, 111110, ... is infinite and prefix-free. So is the set of C-style (null-terminated) strings.

The hypotheses are encoded in a prefix-free way: no hypothesis is a prefix of another hypothesis.

In particular, this eliminates your example: if both strings of length 1 ("0" and "1") are valid hypotheses, then no string of longer length can be a valid hypothesis (then we can take Omega=1 and assign a probability of 1/2 to each). Or we could have "0", "10", and "110" be valid hypotheses; then we can take Omega = 7/8 and assign probabilities of 4/7, 2/7 and 1/7 to these.

In general, the prefix-free condition ensures that the sum over all hypotheses converges, by Kraft's inequality, which is the real heavy lifter here; the constant is merely there to make the sum over all hypotheses converge to 1 instead.

You might imagine that hypotheses are required to be grammatically correct English sentences (I guess encoded in ASCII or something). In that case, no hypothesis is a prefix of another, because each hypothesis ends with a period.

Sorry, I mean that we do this for each value of n. So, given a hypothesis H, it is assigned a probability of 2^-length(H)/Omega.

If we assign a value of 2^-n to each hypothesis (that is, each halting program) of length n, then the total value summed over all hypotheses is Chaitin's Omega.

Thus, if we assign a value of 2^-n/Omega to each hypothesis, the total is 1, and this gives us a valid probability distribution.

Did you not previously state that one should learn about the problem as much as one can before coming to a conclusion, lest one falls prey to the confirmation bias? Should one learn about the problem fully before making a decision only when one doesn't suspect to be biased?

of course not

Use a prefix-free encoding for the hypotheses. There's not 2^n hypotheses of length n: Some of the length-n bitstrings are incomplete and you'd need to add more bits in order to get a hypothesis; others are actually a length <n hypothesis plus some gibberish on the end.

Then the sum of the probabilities of all programs of all lengths combined is 1.0. After excluding the programs that don't halt, the normalization constant is Chaitin's Omega.