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The first requirement of their no-go theorem ("an agent can be certain that a given proposition holds whenever the quantum-mechanical Born rule assigns probabil
by johncolanduoni 8y ago
The first requirement of their no-go theorem ("an agent can be certain that a given proposition holds whenever the quantum-mechanical Born rule assigns probability-1 to it") is interesting, considering the fact that "happens with probability one" is no longer synonymous with "always happens" when you have a probability space with infinite cardinality. Hence the distinction between the technical terms "almost always" and "always" in probability theory.
I haven't finished reading the paper, but does anyone who has already encountered it know if this objection is covered (either in the paper itself or in later discussion)?
- mlthoughts2018 8y agoI don’t believe it was raised anywhere, also just one harp on what you said: it’s not just having infinite cardinality but also having an appropriate probability measure on the space. For example, the positive integers have infinite cardinality, but you can put probability distributions on that space for which there are no non-empty subspaces with probability zero, like (1/2)^k. However I don’t think this issue would actually matter, because from the observer’s point of view in such a situation the probability of discovering you are in one of the Everett branches where the probability-1 event did not occur is exactly 0, so in terms of forming beliefs, you’d assign zero probability to that outcome. Think of it this way, if I create a random continuous function on [0, 1], the probability that it is also differentiable is 0, even though there are tons of familiar examples like x^2 or sin(x). So if I offer you a wager of whether the randomly generated function is differentiable, your belief ought to be that it always is not, even though the technical term would be “almost always” in formal measure theoretic probability theory.