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potato comments on Does Probability Theory Require Deductive or Merely Boolean Omniscience? - Less Wrong Discussion
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Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two. Consider a program that iterates over all possible values of a, b, c, n looking for counterexamples for FLT, then if it finds one, calls a subroutine that eventually prints out X (where X is your current observation). In order to do Solomonoff induction, you need to query a halting oracle on this program. But knowing whether this program halts or not is equivalent to knowing whether FLT is true or false.
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.