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It's arguable that there's no such thing as a probability measure over the reals, because Solomonoff induction only works over computable programs, and the real
by Vecr 3y ago
It's arguable that there's no such thing as a probability measure over the reals, because Solomonoff induction only works over computable programs, and the reals (in the sense needed) are not computable.
- eigenket 3y agoI think such an argument would need quite a lot more work, the lack of Solomonoff induction doesn't mean we don't have probability theory.
- Vecr 3y agoNo, I mean even if you had (perfect, non-approximated) Solomonoff induction, you could only generate probabilities for computable "theories" (programs that predict all your past and future input), but I suppose it's possible that the impossibility proofs actually depend in some way on Aleph 1, so you would need it for consistency.