As originally formulated by Nozick, Omega is not necessarily omniscient and does not necessarily have anything like divine foreknowledge. All that is said about this is that you have “enormous confidence” in Omega’s power to predict your choices, and that this being has “often correctly predicted your choices in the past (and has never, as far as you know made an incorrect prediction about your choices)”, and that the being has “often correctly predicted the choices of other people, many who are similar to you”. So, all I really know about Omega is that it has a really good track record.
So, nothing in Nozick rules out the possibility of the outcome “b” or “c” listed above.
At the time that you make your choice, Omega has already irrevocably either put $1M in box 2 or put nothing in box 2
If Omega has put $1M in box 2, your payoff will be $1M if you 1-box or 1.001M if you 2-box.
If Omega has put nothing in box 2, your payoff will be $0 if you 1-box or $1K if you 2-box.
So, whatever Omega has already done, you are better off 2-boxing. And, your choice now cannot change what Omega has already done.
So, you are better off 2-boxing.
So, basically, I agree with your assessment that “two-boxers believe that all 4 are possible” (or at least I believe that all 4 are possible). Why do I believe that all 4 are possible? Because nothing in the problem statement says otherwise.
ETA:
Also, I agree with your assessment that “one-boxers do not believe b and c are possible because Omega is cheating or a perfect predictor (same thing)”. But, in thinking this way, one-boxers are reading something into the problem beyond what is actually stated or implied by Nozick.
I’m a two-boxer. My rationale is:
As originally formulated by Nozick, Omega is not necessarily omniscient and does not necessarily have anything like divine foreknowledge. All that is said about this is that you have “enormous confidence” in Omega’s power to predict your choices, and that this being has “often correctly predicted your choices in the past (and has never, as far as you know made an incorrect prediction about your choices)”, and that the being has “often correctly predicted the choices of other people, many who are similar to you”. So, all I really know about Omega is that it has a really good track record.
So, nothing in Nozick rules out the possibility of the outcome “b” or “c” listed above.
At the time that you make your choice, Omega has already irrevocably either put $1M in box 2 or put nothing in box 2
If Omega has put $1M in box 2, your payoff will be $1M if you 1-box or 1.001M if you 2-box.
If Omega has put nothing in box 2, your payoff will be $0 if you 1-box or $1K if you 2-box.
So, whatever Omega has already done, you are better off 2-boxing. And, your choice now cannot change what Omega has already done.
So, you are better off 2-boxing.
So, basically, I agree with your assessment that “two-boxers believe that all 4 are possible” (or at least I believe that all 4 are possible). Why do I believe that all 4 are possible? Because nothing in the problem statement says otherwise.
ETA:
Also, I agree with your assessment that “one-boxers do not believe b and c are possible because Omega is cheating or a perfect predictor (same thing)”. But, in thinking this way, one-boxers are reading something into the problem beyond what is actually stated or implied by Nozick.