At 1/19/04, Hal Finney wrote:However, here is an alternate formulation of my argument which seems to be roughly equivalent and which avoids this objection: create a random program tape by flipping a coin for each bit. Now the probability that you created the first program above is 1/2^100, and for the second, 1/2^120, so the first program is 2^20 times more probable than the second.
That's an interesting idea, but I don't know what to make of it. All it does is create a conflict of intuition which I don't know how to resolve. On the one hand, the following argument seems to make sense: consider an infinite sequence of random bits. The probability that the sequence begins with "1" is .5. The probability that it begins with "01" is .25. Therefore, in the uncountably infinite set of all possible infinite bit-strings, those that begin with "1" are twice as common as those that begin with "01". However, this is in direct conflict with the intuition which says that, since there are uncountably many infinite bit-strings that begin with "1", and uncountably many that begin with "01", the two types of strings are equally as common. How can we resolve this conflict?
I haven't studied measure theory, but from reading definitions and seeing discussions my understanding is that it's about functions that assign real numbers to collections of subsets (defined by 'sigma algebras') of infinite sets. As applied to probability theory, it allows you to define a notion of probability on a set with an infinite number of members. Again, this would involve assigning probabilities to *subsets* of this infinite set, not to every member of the infinite set--for example, if you are dealing with the set of real numbers between 0 and 1, then although each individual real number could not have a finite probability (since this would not be compatible with the idea that the total probability must be 1), perhaps each finite nonzero interval (say, 0.5 - 0.8) would have a finite probability. In a similar way, if you were looking at the set of all possible infinite bit-strings, although each individual string might not get a probability, you might have a measure that can tell you the probability of getting a member of the subset "strings beginning with 1" vs. the probability of getting a member of the subset "strings beginning with 01". Some references on measure theory that may be helpful:
http://en2.wikipedia.org/wiki/Measure_theory http://en2.wikipedia.org/wiki/Sigma_algebra http://en2.wikipedia.org/wiki/Probability_axioms http://mathworld.wolfram.com/Measure.html http://mathworld.wolfram.com/ProbabilityMeasure.html
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