Doesn’t the non-monotonous convergence of the existential statements pose a problem for practical application? The process eventually converges to truth, even in finite time, but before it does, the estimates move in the wrong direction.
As for Goedelian limitations: is it correct to suppose that the statements which are undecidable in an axiomatic system would have oscillating (i.e. non-convergent) probabilities in this probabilistic system? Is it guaranteed that contradictions don’t arise even here? (By a contradicition, I mean a statement X whose probability converges at p while simultaneously the probability of non-X converge at anything different from 1-p. Of course this can be prevented by implementing an updating rule which, after each update, ensures that the probabilities of X and non-X add up to 1. But this doesn’t seem particularly elegant: it would be analogous to declaring non-X as non-theorem after proving X in a formal system.)
Doesn’t the non-monotonous convergence of the existential statements pose a problem for practical application? The process eventually converges to truth, even in finite time, but before it does, the estimates move in the wrong direction.
This is right. Actually “forall” statements can converge non-monotonously too, if they’re false. I believe Goedel’s theorem implies you can’t make a monotonously converging algorithm.
is it correct to suppose that the statements which are undecidable in an axiomatic system would have oscillating (i.e. non-convergent) probabilities in this probabilistic system?
Hmm. This seems to be wrong because my construction doesn’t use any single axiomatic system.
ETA: I thought a little more and it seems we can’t get rid of “oscillating” statements. Thanks! My whole post is wrong :-)
Doesn’t the non-monotonous convergence of the existential statements pose a problem for practical application? The process eventually converges to truth, even in finite time, but before it does, the estimates move in the wrong direction.
As for Goedelian limitations: is it correct to suppose that the statements which are undecidable in an axiomatic system would have oscillating (i.e. non-convergent) probabilities in this probabilistic system? Is it guaranteed that contradictions don’t arise even here? (By a contradicition, I mean a statement X whose probability converges at p while simultaneously the probability of non-X converge at anything different from 1-p. Of course this can be prevented by implementing an updating rule which, after each update, ensures that the probabilities of X and non-X add up to 1. But this doesn’t seem particularly elegant: it would be analogous to declaring non-X as non-theorem after proving X in a formal system.)
This is right. Actually “forall” statements can converge non-monotonously too, if they’re false. I believe Goedel’s theorem implies you can’t make a monotonously converging algorithm.
Hmm. This seems to be wrong because my construction doesn’t use any single axiomatic system.
ETA: I thought a little more and it seems we can’t get rid of “oscillating” statements. Thanks! My whole post is wrong :-)