I thought about your questions some more, and stumbled upon a perspective that makes them all meaningful—yes, even the one about defining the real numbers. You have to imagine yourself living in a sort of “Solomonoff multiverse” that runs a weighted mix of all possible programs, and act as if to maximize your expected utility over that whole multiverse. Never mind “truth” or “degrees of belief” at all! If Omega comes to you and asks whether an inaccessible region of space contains galaxies or unicorns, bravely answer “galaxies” because it wins you more cookies weighted by universe-weight—simpler programs have more of it. This seems to be the coherent position that many commenters seem to be groping toward…
I thought about your questions some more, and stumbled upon a perspective that makes them all meaningful—yes, even the one about defining the real numbers. You have to imagine yourself living in a sort of “Solomonoff multiverse” that runs a weighted mix of all possible programs, and act as if to maximize your expected utility over that whole multiverse. Never mind “truth” or “degrees of belief” at all! If Omega comes to you and asks whether an inaccessible region of space contains galaxies or unicorns, bravely answer “galaxies” because it wins you more cookies weighted by universe-weight—simpler programs have more of it. This seems to be the coherent position that many commenters seem to be groping toward…