Yep, you’re confused. You can’t possibly assign equal probability to every program in an infinite space, because then the sum of all probabilities will diverge (go to infinity), and you need it to sum up to 1.
Only if you insist on assigning non-zero probability to each individual program.
If you assign an equal probability to every program, and that probability is zero, then the sum, the total probability, is zero and. If you assign an equal nonzero probability, the probability is infinite.
If you want to have a valid probability distribution, you can’t assign equal probabilities.
Well, if you’re willing to relax the third axiom of probability to finite additivity you can. Alternatively, use an uncountable state space, e.g., the space of all ultra-filters on Turing machines. Also, while we’re on the subject, there are numerous types of quasi-probability distributions worth considering.
If you assign an equal probability to every program, and that probability is zero, then the sum, the total probability, is zero and. If you assign an equal nonzero probability, the probability is infinite.
If you want to have a valid probability distribution, you can’t assign equal probabilities.
Well, if you’re willing to relax the third axiom of probability to finite additivity you can. Alternatively, use an uncountable state space, e.g., the space of all ultra-filters on Turing machines. Also, while we’re on the subject, there are numerous types of quasi-probability distributions worth considering.
Ah, fair enough. I had been assuming only real probabilities.