It is not true that if someone has the ability to harm n people, then they also necessarily have the ability to harm exactly m of those people for any m<n, so it isn’t clear that P(there is someone who has the ability to harm n people) monotonically decreases as n increases. Unless you meant at least n people, in which case that’s true, but still irrelevent since it doesn’t even establish that this probability approaches a limit of 0, much less that it does so at any particular rate.
Compare it to an artifical lottery where we force our computer to draw a number between 0 and 10^10^27 (deliberately chosen larger than the ‘practical limit’ referred to in the post). I think we can justly assign the claim [The number 3 will be the winning lottery ticket] a probability of 1/10^10^27.
The probability that a number chosen randomly from the uniform distribution on integers from 0 to 10^10^27 is 3 is indeed 1/10^10^27, but I wouldn’t count that as an empirical hypothesis. Given any particular mechanism for producing integers that you are very confident implements a uniform distribution on the integers from 0 to 10^10^27, the probability you should assign to it producing 3 is still much higher than 1/10^10^27.
Or in more Bayesian terms—which evidence E is more likely if this stranger can harm 3↑↑↑3 people than if it can harm, say, only 10^100 people?
The stranger says so, and has established some credibility by providing very strong evidence for other a priori very implausible claims that they have made.
Any suggestions along the lines of ‘present 3↑↑↑3 people and punch them all in the nose’ justifies having a prior of 1/3↑↑↑3 for this event, since showing that many people really is evidence with that likelihood ratio.
That’s not evidence that is physically possible to present to a human, and I don’t see why you say its likelihood ratio is around 1:3↑↑↑3.
I think I will try writing my reply as a full post, this discussion is getting longer than is easy to fit as a set of replies. You are right that my above reply has some serious flaws.
It is not true that if someone has the ability to harm n people, then they also necessarily have the ability to harm exactly m of those people for any m<n, so it isn’t clear that P(there is someone who has the ability to harm n people) monotonically decreases as n increases. Unless you meant at least n people, in which case that’s true, but still irrelevent since it doesn’t even establish that this probability approaches a limit of 0, much less that it does so at any particular rate.
The probability that a number chosen randomly from the uniform distribution on integers from 0 to 10^10^27 is 3 is indeed 1/10^10^27, but I wouldn’t count that as an empirical hypothesis. Given any particular mechanism for producing integers that you are very confident implements a uniform distribution on the integers from 0 to 10^10^27, the probability you should assign to it producing 3 is still much higher than 1/10^10^27.
The stranger says so, and has established some credibility by providing very strong evidence for other a priori very implausible claims that they have made.
That’s not evidence that is physically possible to present to a human, and I don’t see why you say its likelihood ratio is around 1:3↑↑↑3.
I think I will try writing my reply as a full post, this discussion is getting longer than is easy to fit as a set of replies. You are right that my above reply has some serious flaws.