But, if “all mathematical structures are real,” and possible universes, that must include structures like the diophantine equations isomorphic to Chaitin’s Omega, and other noncomputable stuff, right? Can algorithmic probability tell me what mathematical structure generated a string, when some of the possible mathematical structures are not computable?
Can algorithmic probability tell me what mathematical structure generated a string, when some of the possible mathematical structures are not computable?
Presumably you’ll only ever attempt to infer from a finite prefix of such a string, which is guaranteed to have a computable description. (Worst case scenario: “the string whose first character is L, whose second character is e, …”).
But, we’re trying to infer the actual generator, right? If, big-worldily, the actual generator is in some set of incomputible generators, it doesn’t help us at all to come up with an n-parameter generating program for the first n bits—although no computable predictor can do better. If the set of possible generators is, itself, incomputable, how do we set a probability distribution over it?
No. Why would it?
Caveat: I can not math good.
But, if “all mathematical structures are real,” and possible universes, that must include structures like the diophantine equations isomorphic to Chaitin’s Omega, and other noncomputable stuff, right? Can algorithmic probability tell me what mathematical structure generated a string, when some of the possible mathematical structures are not computable?
Presumably you’ll only ever attempt to infer from a finite prefix of such a string, which is guaranteed to have a computable description. (Worst case scenario: “the string whose first character is L, whose second character is e, …”).
But, we’re trying to infer the actual generator, right? If, big-worldily, the actual generator is in some set of incomputible generators, it doesn’t help us at all to come up with an n-parameter generating program for the first n bits—although no computable predictor can do better. If the set of possible generators is, itself, incomputable, how do we set a probability distribution over it?
Ah, good point. Yes, that’s probably a safe assumption for practical purposes.