Is that responding to the “You can’t possibly assign equal probability to every program in an infinite space” part of the parent comment? If you assign equal probability to every program in an infinite space, then (assuming “probability” refers to a real number, and not some other mathematical object such as an infinitesimal) either that equal probability is non-zero, in which case the sum is infinite, or the equal probability is zero, in which case the sum is zero. Remember, if all the probabilities are equal, then either all of them are equal to zero, or none are, and it’s generally agreed that an infinite number of zeros add up to zero.
and it’s generally agreed that an infinite number of zeros add up to zero.
Actually 0·∞ is considered an indeterminate form. For example the interval from 0 to 1 has length 1 even though it’s composed of infinitely many points each of which has length 0.
See my reply to asr for links to more technical explanations.
Is that responding to the “You can’t possibly assign equal probability to every program in an infinite space” part of the parent comment? If you assign equal probability to every program in an infinite space, then (assuming “probability” refers to a real number, and not some other mathematical object such as an infinitesimal) either that equal probability is non-zero, in which case the sum is infinite, or the equal probability is zero, in which case the sum is zero. Remember, if all the probabilities are equal, then either all of them are equal to zero, or none are, and it’s generally agreed that an infinite number of zeros add up to zero.
Actually 0·∞ is considered an indeterminate form. For example the interval from 0 to 1 has length 1 even though it’s composed of infinitely many points each of which has length 0.
See my reply to asr for links to more technical explanations.