On Sat, Jan 24, 2004 at 12:21:40PM -0800, Eric Hawthorne wrote: > Can you explain briefly why the choice of measure is subjective? I > haven't read any of the > books you mentioned (will try to get to them) but am familiar with > computability theory > and decision theory.
Since you do not mention that you're familiar with the theory of algorithmic complexity, I suggest that you read the first book on that list ASAP. The following response might not make sense until you do. Basically, all of the sensible proposed measures are based on the universal distribution, which assigns a larger probabilities to strings that have lower algorithmic complexities. However there's actually an infinite class of universal distributions, one for each universal Turing machine, and there's no objective criteria for determining which one should be used. Another problem is that using the universal distribution forces you to assume that non-computable universes do not exist. If one does not want to make this assumption, then a more dominant measure need to be used (for example, based on a TM with an oracle for the halting problem or the complexity of a string's logical definition) but then there are even more measures to choose from (how high up the computability hierarchy do you go? how high up the set theoretic hierarchy?). Now suppose that two people, Alice and Bob, somehow agree that a measure M is the objectively correct measure, but Bob insists on using measure M' in making decisions. He says "So what if universe A has a bigger measure than universe B according to M? I just care more about what happens in universe B than universe A, so I'll use M' which assigns a bigger measure to universe B." What can Alice say to Bob to convince him that he is not being rational? I don't see what the answer could be.
