I think the reason is sounds so odd is: how the hell is Omega calculating what your answer would have been if 1=0?
If what Omega is really calculating is what you would have done if you were merely told something equivalent to 1=0, then sure, paying up can make sense.
It seems to me that the relevant difference between “1=0” and “the billionth digit of pi is even” is that the latter statement has a really long disproof, but there might be a much shorter proof of what the agent would do if that statement were true. Or at least I imagine Omega to be doing the same sort of proof-theoretic counterfactual reasoning that’s described in the post. Though maybe there’s some better formalization of Counterfactual Mugging with a logical coin that we haven’t found...
Even if you’re cutting off Omega’s proofs at some length, there are plenty of math problems that people can’t do that are shorter than high-probability predictions that people will or won’t pay up. Certainly when I imagine the problem, I imagine it in the form of predicting someone who’s been told that the trillionth digit of pi is even and then paying out to that person depending on their counterfactual actions.
Of course, that leads to odd situations when the agent being predicted can do the math problem, but Omega still says “no bro, trust me, the trillionth digit of pi really is even.” But an agent who can do the math will still give Omega the money because decision theory, so does it really matter?
If you’re proposing to treat Omega’s words as just observational evidence that isn’t connected to math and could turn out one way or the other with probability 50%, I suppose the existing formalizations of UDT already cover such problems. But how does the agent assign probability 50% to a particular math statement made by Omega? If it’s more complicated than “the trillionth digit of pi is even”, then the agent needs some sort of logical prior over inconsistent theories to calculate the probabilities, and needs to be smart enough to treat these probabilities updatelessly, which brings us back to the questions asked at the beginning of my post… Or maybe I’m missing something, can you specify your proposal in more detail?
Well, I was thinking more in terms of a logical prior over single statements, see my favorite here.
But yeah I guess I was missing the point of the problem.
Also: suppose Omega comes up to you and says “If 1=0 was true I would have given you billion dollars if and only if you would give me 100 dollars if 1=1 was true. 1=1 is true, so can you spare $100?” Does this sound trustworthy? Frankly not, it feels like there’s a principle of explosion problem that insists that Omega would have given you all possible amounts of money at once if 1=0 was true.
A formulation that avoids the principle of explosion is “I used some process that I cannot prove the outcome of to pick a digit of pi. If that digit of pi was odd I would have given you a billion dollars iff [etc].”
Hm, yeah, that sounds really odd.
I think the reason is sounds so odd is: how the hell is Omega calculating what your answer would have been if 1=0?
If what Omega is really calculating is what you would have done if you were merely told something equivalent to 1=0, then sure, paying up can make sense.
It seems to me that the relevant difference between “1=0” and “the billionth digit of pi is even” is that the latter statement has a really long disproof, but there might be a much shorter proof of what the agent would do if that statement were true. Or at least I imagine Omega to be doing the same sort of proof-theoretic counterfactual reasoning that’s described in the post. Though maybe there’s some better formalization of Counterfactual Mugging with a logical coin that we haven’t found...
Even if you’re cutting off Omega’s proofs at some length, there are plenty of math problems that people can’t do that are shorter than high-probability predictions that people will or won’t pay up. Certainly when I imagine the problem, I imagine it in the form of predicting someone who’s been told that the trillionth digit of pi is even and then paying out to that person depending on their counterfactual actions.
Of course, that leads to odd situations when the agent being predicted can do the math problem, but Omega still says “no bro, trust me, the trillionth digit of pi really is even.” But an agent who can do the math will still give Omega the money because decision theory, so does it really matter?
If you’re proposing to treat Omega’s words as just observational evidence that isn’t connected to math and could turn out one way or the other with probability 50%, I suppose the existing formalizations of UDT already cover such problems. But how does the agent assign probability 50% to a particular math statement made by Omega? If it’s more complicated than “the trillionth digit of pi is even”, then the agent needs some sort of logical prior over inconsistent theories to calculate the probabilities, and needs to be smart enough to treat these probabilities updatelessly, which brings us back to the questions asked at the beginning of my post… Or maybe I’m missing something, can you specify your proposal in more detail?
Well, I was thinking more in terms of a logical prior over single statements, see my favorite here.
But yeah I guess I was missing the point of the problem.
Also: suppose Omega comes up to you and says “If 1=0 was true I would have given you billion dollars if and only if you would give me 100 dollars if 1=1 was true. 1=1 is true, so can you spare $100?” Does this sound trustworthy? Frankly not, it feels like there’s a principle of explosion problem that insists that Omega would have given you all possible amounts of money at once if 1=0 was true.
A formulation that avoids the principle of explosion is “I used some process that I cannot prove the outcome of to pick a digit of pi. If that digit of pi was odd I would have given you a billion dollars iff [etc].”