“The experiment being repeated sufficiently often might be considered a reasonably mild restriction; in particular, it is a given if the universe is large enough that everything which appears once appears many times.”
Why is that a given? The set of integers is very large, but the number 3 only appears once in it.
I think the relevant difference is that, in the set of integers, each element is strictly more complex than the previous one, but in the universe, you can probably upper bound the complexity (that’s what I’m assuming, anyway). So eventually stuff should repeat, and then anything that has a nonzero probability of appearing will appear arbitrarily often as you increase the size. For example, if there’s an upper bound to the complexity of a planet, then you can only have that many planets until you get a repeat.
That doesn’t seem to follow, actually. You could easily have a very large universe that’s almost entirely empty space (which does “repeat”), plus a moderate amount of structures that only appear once each.
And as a separate argument, plenty of processes are irreversible in practice. For instance, consider a universe where there’s a “big bang” event at the start of time, like an ordinary explosion. I’d expect that universe to never return to that original intensely-exploding state, because the results of explosions don’t go backwards in time, right?
That doesn’t seem to follow, actually. You could easily have a very large universe that’s almost entirely empty space (which does “repeat”), plus a moderate amount of structures that only appear once each.
Yeah, nonemptiness was meant to be part of the assumption in the phrase you quoted.
And as a separate argument, plenty of processes are irreversible in practice. For instance, consider a universe where there’s a “big bang” event at the start of time, like an ordinary explosion. I’d expect that universe to never return to that original intensely-exploding state, because the results of explosions don’t go backwards in time, right?
We’re getting into territory where I don’t feel qualified to argue – although it seems like that objection only applies to some very specific things, and probably not to most Sleeping Beauty like scenarios.
Thanks for the mention, I had never heard of that concept before.
I have strong reflexes of revulsion against this idea that everything must reoccur (aren’t plenty of processes irreversible in our world?), but it’s getting too off-topic for the original article, and I need to think more about this.
“The experiment being repeated sufficiently often might be considered a reasonably mild restriction; in particular, it is a given if the universe is large enough that everything which appears once appears many times.”
Why is that a given? The set of integers is very large, but the number 3 only appears once in it.
I think the relevant difference is that, in the set of integers, each element is strictly more complex than the previous one, but in the universe, you can probably upper bound the complexity (that’s what I’m assuming, anyway). So eventually stuff should repeat, and then anything that has a nonzero probability of appearing will appear arbitrarily often as you increase the size. For example, if there’s an upper bound to the complexity of a planet, then you can only have that many planets until you get a repeat.
That doesn’t seem to follow, actually. You could easily have a very large universe that’s almost entirely empty space (which does “repeat”), plus a moderate amount of structures that only appear once each.
And as a separate argument, plenty of processes are irreversible in practice. For instance, consider a universe where there’s a “big bang” event at the start of time, like an ordinary explosion. I’d expect that universe to never return to that original intensely-exploding state, because the results of explosions don’t go backwards in time, right?
Yeah, nonemptiness was meant to be part of the assumption in the phrase you quoted.
We’re getting into territory where I don’t feel qualified to argue – although it seems like that objection only applies to some very specific things, and probably not to most Sleeping Beauty like scenarios.
Not by algorithmic complexity. The integer consisting of a million 3s in a row is quite compressible.
But by number of bits, which is what you need to avoid repetition.
The typical answer is that this is a result of the Poincaré recurrence theorem
Thanks for the mention, I had never heard of that concept before.
I have strong reflexes of revulsion against this idea that everything must reoccur (aren’t plenty of processes irreversible in our world?), but it’s getting too off-topic for the original article, and I need to think more about this.