What stops the total number of humans from being infinite is the thermodynamic death of the universe.
I’m not a cosmologist (so please do correct me if I’m mistaken here!) but I think the universe’s lifespan can be divided up into epochs, and whilst Boltzmann Brains can exist within any epoch, humans can only exist in a limited few: too early and the universe is too violent and chaotic to support us (plus probably not all the elements we’re made from have been produced yet); too late and there isn’t sufficient negentropy left to support us.
We’re currently in the ‘stellar’ epoch—but I think we’re already past the peak of stellar formation: the rate at which new stars are forming is slowing down.
The ‘black hole’ epoch will come afterwards, once the stars have all burned themselves out and there’s no fusible material left to form new ones, but that’s probably fine: we can probably extract energy from black holes (and from whatever residual heat there remains in stellar remnants like brown dwarves etc.)
But, after even that, there might be an infinitely long ‘de Sitter’ epoch, after the last black hole has evapourated into nothing, the last stellar remnant has fully cooled, and there are simply no remaining sources of free energy anywhere. Boltzmann Brains could exist within this epoch (they could coalesce spontaneously from the ever-present quantum vacuum fluctuations) but regular humans, even digitally simulated ones, could not.
The idea being that, since Earth is possible, a spacially infinite universe could have an infinite number of Earths with humans on them, even if the Boltzmann Brain epoch is longer than the stellar epoch. It’s hard to compare infinities.
Partially agree. If the universe were both infinite in size and contained infinite negentropy (or, if you prefer, infinite matter and energy) then sure, I’d agree that there would likely be vastly more human brains than Boltzmann brains.
However, if the universe were infinite in size but didn’t contain an infinite amount of negentropy (for example, if the universe started with some fixed amount of negentropy -like all the matter and energy present at the Big Bang- and then became infinite in size by inflating for an infinite amount of time, but after the initial Big Bang no additional matter/energy was added) then I’d say the infinite size alone wouldn’t be sufficient to make human brains more probable.
I don’t know enough about cosmology to know whether there’s any sort of consensus on whether the amount of negentropy actually is finite or infinite. I was (as you probably deduced from my previous comment!) just assuming a universe that’s infinite in size (both spatially and temporally) but not in negentropy (hence the maximum-entropy de Sitter ‘heat death’ epoch).
“It’s hard to compare infinities”
True, but that doesn’t mean we can’t make any comparisons between them! For example, I think we can say “If the universe is infinite in size and contains an infinite amount of negentropy/matter, then the “universe size” infinity must be a larger infinity than the “amount of matter” infinity, otherwise we would observe infinite density throughout the universe”.
There are purer examples: Cantor’s infinite series maths, where by matching-up elements in sets you can compare not only the relative sizes of infinite sets but also the “size classes” of types of infinite set. So eg. you can compare the size of the set of positive even numbers (let’s call that Set A) to the size of the set of positive integers (Set B) by saying “if we match up the 2 from Set A with the 2 from Set B, the 4 from Set A with the 4 from Set B, the 6 from Set A with the 6 from Set B, and so on all the way to infinity, we’d still have 50% of the numbers in Set B left over, therefore Set A is 50% the size of Set B” and you can say “If Set C is ‘the set of all the numbers between 0.0 and 1.0’ there’s no way to match-up the numbers in Set C with the numbers in Set B, therefore Set C is a bigger ‘size class’ than Set B”.
(Confession: pretty sure Cantor didn’t call this sort of thing a “size class” but I have no idea what it is called, so I made up the “size class” term..)
Can we make use of this in cosmology? I think so: my “infinite density” argument above basically works like this. More pertinently, if we could assess the probability of a human brain existing (per unit spacetime, per unit negentropy) as (say) 10^-50 and the probability of a Boltzmann Brain existing as 10^-52, we would be able to say “There are around 100 times more human brains than Boltzmann brains”, even though both could be infinite if the spacetime and negentropy were infinite.
What stops the total number of humans from being infinite is the thermodynamic death of the universe.
I’m not a cosmologist (so please do correct me if I’m mistaken here!) but I think the universe’s lifespan can be divided up into epochs, and whilst Boltzmann Brains can exist within any epoch, humans can only exist in a limited few: too early and the universe is too violent and chaotic to support us (plus probably not all the elements we’re made from have been produced yet); too late and there isn’t sufficient negentropy left to support us.
We’re currently in the ‘stellar’ epoch—but I think we’re already past the peak of stellar formation: the rate at which new stars are forming is slowing down.
The ‘black hole’ epoch will come afterwards, once the stars have all burned themselves out and there’s no fusible material left to form new ones, but that’s probably fine: we can probably extract energy from black holes (and from whatever residual heat there remains in stellar remnants like brown dwarves etc.)
But, after even that, there might be an infinitely long ‘de Sitter’ epoch, after the last black hole has evapourated into nothing, the last stellar remnant has fully cooled, and there are simply no remaining sources of free energy anywhere. Boltzmann Brains could exist within this epoch (they could coalesce spontaneously from the ever-present quantum vacuum fluctuations) but regular humans, even digitally simulated ones, could not.
The idea being that, since Earth is possible, a spacially infinite universe could have an infinite number of Earths with humans on them, even if the Boltzmann Brain epoch is longer than the stellar epoch. It’s hard to compare infinities.
Partially agree. If the universe were both infinite in size and contained infinite negentropy (or, if you prefer, infinite matter and energy) then sure, I’d agree that there would likely be vastly more human brains than Boltzmann brains.
However, if the universe were infinite in size but didn’t contain an infinite amount of negentropy (for example, if the universe started with some fixed amount of negentropy -like all the matter and energy present at the Big Bang- and then became infinite in size by inflating for an infinite amount of time, but after the initial Big Bang no additional matter/energy was added) then I’d say the infinite size alone wouldn’t be sufficient to make human brains more probable.
I don’t know enough about cosmology to know whether there’s any sort of consensus on whether the amount of negentropy actually is finite or infinite. I was (as you probably deduced from my previous comment!) just assuming a universe that’s infinite in size (both spatially and temporally) but not in negentropy (hence the maximum-entropy de Sitter ‘heat death’ epoch).
True, but that doesn’t mean we can’t make any comparisons between them! For example, I think we can say “If the universe is infinite in size and contains an infinite amount of negentropy/matter, then the “universe size” infinity must be a larger infinity than the “amount of matter” infinity, otherwise we would observe infinite density throughout the universe”.
There are purer examples: Cantor’s infinite series maths, where by matching-up elements in sets you can compare not only the relative sizes of infinite sets but also the “size classes” of types of infinite set. So eg. you can compare the size of the set of positive even numbers (let’s call that Set A) to the size of the set of positive integers (Set B) by saying “if we match up the 2 from Set A with the 2 from Set B, the 4 from Set A with the 4 from Set B, the 6 from Set A with the 6 from Set B, and so on all the way to infinity, we’d still have 50% of the numbers in Set B left over, therefore Set A is 50% the size of Set B” and you can say “If Set C is ‘the set of all the numbers between 0.0 and 1.0’ there’s no way to match-up the numbers in Set C with the numbers in Set B, therefore Set C is a bigger ‘size class’ than Set B”.
(Confession: pretty sure Cantor didn’t call this sort of thing a “size class” but I have no idea what it is called, so I made up the “size class” term..)
Can we make use of this in cosmology? I think so: my “infinite density” argument above basically works like this. More pertinently, if we could assess the probability of a human brain existing (per unit spacetime, per unit negentropy) as (say) 10^-50 and the probability of a Boltzmann Brain existing as 10^-52, we would be able to say “There are around 100 times more human brains than Boltzmann brains”, even though both could be infinite if the spacetime and negentropy were infinite.
“Cardinality” is the math term you’re looking for.