If we choose P, then X “exists” iff P appears somewhere in (a symbolic representation of) our universe. If we choose P2, then X “exists” iff P2 appears somewhere in (a symbolic representation of) our universe.
There is no reason why these two propositions should be equivalent. So whether we choose P or P2 may make a difference to whether or not X “exists”.
As I understand this, the two propositions are equivalent. We do not arbitrarily pick P or P2 (we assume that one is picked and is picked consistently). What this means is that the G(TM) will also pick P or P2 consistently. If G(TM) outputs P, then it would never output P2 and vice versa. Only one of the members of the equivalence class become the shortest program, and that member represents the entire class everytime the class is invoked. So the shortest program will be computed by G consistently when simulating our universe.
My more fundamental objection is that it seems perfectly obvious to me that whether P appears in G(Q) has nothing whatever to do with whether X exists, because there is no reason why the universe should contain programs implementing all its objects.)
Yes, I agree that the universe can be simulated (or, more precisely, my guess is that it can be, the available scientific evidence suggests that it probably can be, and it’s a convenient working hypothesis).
I’m afraid I don’t at all understand your argument here. “If G(TM) outputs P, then it would never output P2 and vice versa”—I have no inkling why that should be true. Why do you believe that G(TM) cannot output both of them on different occasions?
As I understand this, the two propositions are equivalent. We do not arbitrarily pick P or P2 (we assume that one is picked and is picked consistently). What this means is that the G(TM) will also pick P or P2 consistently. If G(TM) outputs P, then it would never output P2 and vice versa. Only one of the members of the equivalence class become the shortest program, and that member represents the entire class everytime the class is invoked. So the shortest program will be computed by G consistently when simulating our universe.
Do you agree that the universe can be simulated?
(Sorry about the long delay in replying.)
Yes, I agree that the universe can be simulated (or, more precisely, my guess is that it can be, the available scientific evidence suggests that it probably can be, and it’s a convenient working hypothesis).
I’m afraid I don’t at all understand your argument here. “If G(TM) outputs P, then it would never output P2 and vice versa”—I have no inkling why that should be true. Why do you believe that G(TM) cannot output both of them on different occasions?