potato comments on Does Probability Theory Require Deductive or Merely Boolean Omniscience? - Less Wrong

4 Post author: potato 03 August 2015 06:54AM

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Comment author: potato 03 August 2015 11:50:44PM *  0 points [-]

Let's forget about the oracle. What about the program that outputs X only if 1 + 1 = 2, and else prints 0? Let's call it A(1,1). The formalism requires that P(X|A(1,1)) = 1, and it requires that P(A(1,1)) = 2 ^-K(A(1,1,)), but does it need to know that "1 + 1 = 2" is somehow proven by A(1,1) printing X?

In either case, you've shown me something that I explicitly doubted before: one can prove any provable theorem if they have access to a Solomonoff agent's distribution, and they know how to make a program that prints X iff theorem S is provable. All they have to do is check the probability the agent assigns to X conditional on that program.