She opens the second box, and goes home with $1,001,000. Why shouldn’t she? Omega’s dead.
No, she goes home with $1,000. She two-boxes, so Omega predicted she’d two-box.
“But Omega is dead, and she sees the $1,000,000.”
Sure, but you can’t have it both ways. You can’t have a perfect predictor and have Irene disprove the prediction.
If Omega’s prediction isn’t perfect, but e.g. 99% accurate, two-boxing on a one-box prediction might be possible, but still isn’t wise, since then Omega will almost certainly have put the $1,000,000 in the box.
Assuming perfect prediction by Omega:
No, she goes home with $1,000. She two-boxes, so Omega predicted she’d two-box.
“But Omega is dead, and she sees the $1,000,000.”
Sure, but you can’t have it both ways. You can’t have a perfect predictor and have Irene disprove the prediction.
If Omega’s prediction isn’t perfect, but e.g. 99% accurate, two-boxing on a one-box prediction might be possible, but still isn’t wise, since then Omega will almost certainly have put the $1,000,000 in the box.