I’m not sure that both these statements can be true at the same time.
If you take the second statement to mean, “There exists an algorithm for Omega satisfying the probabilities for correctness in all cases, and which sometimes outputs the same number as NL, which does not take NL’s number as an input, for any algorithm Player taking NL’s and Omega’s numbers as input,” then this …seems… true.
I haven’t yet seen a comment that proves it, however. In your example, let’s assume that we have some algorithm for NL with some specified probability of outputting a prime number, and some specified probability it will end in 3, and maybe some distribution over magnitude. Then Omega need only have an algorithm that outputs combinations of primeness and 3-endedness such that the probabilities of outcomes are satisfied, and which sometimes produces coincidences.
For some algorithms of NL, this is clearly impossible (e.g. NL always outputs prime, c.f. a Player who always two-boxes). What seems less certain is whether there exists an NL for which Omega can always generate an algorithm (satisfying both 99.9% probabilities) for any algorithm of the Player.
This is to say, what we might have in the statement of the problem is evidence for what sort of algorithm the Natural Lottery runs.
Perhaps what Eliezer means is that the primeness of Omega’s number may be influenced by the primeness NL’s number, but not by which number specifically? Maybe the second statement is meant to suggest something about the likelihood of there being a coincidence?
If you take the second statement to mean, “There exists an algorithm for Omega satisfying the probabilities for correctness in all cases, and which sometimes outputs the same number as NL, which does not take NL’s number as an input, for any algorithm Player taking NL’s and Omega’s numbers as input,” then this …seems… true.
I haven’t yet seen a comment that proves it, however. In your example, let’s assume that we have some algorithm for NL with some specified probability of outputting a prime number, and some specified probability it will end in 3, and maybe some distribution over magnitude. Then Omega need only have an algorithm that outputs combinations of primeness and 3-endedness such that the probabilities of outcomes are satisfied, and which sometimes produces coincidences.
For some algorithms of NL, this is clearly impossible (e.g. NL always outputs prime, c.f. a Player who always two-boxes). What seems less certain is whether there exists an NL for which Omega can always generate an algorithm (satisfying both 99.9% probabilities) for any algorithm of the Player.
This is to say, what we might have in the statement of the problem is evidence for what sort of algorithm the Natural Lottery runs.
Perhaps what Eliezer means is that the primeness of Omega’s number may be influenced by the primeness NL’s number, but not by which number specifically? Maybe the second statement is meant to suggest something about the likelihood of there being a coincidence?