the universe is infinite in the sense that every possible combination of atoms is repeated an infinite number of times (either because the negative curvature of the universe implies the universe is unbounded or because of MWI)
As a side note, I do not like seeing these two possible causes of this belief be put in the same bucket. I think they are quite different in an important respect: while it seems MWI would indeed imply this conclusion, negative curvature alone certainly would not. A universe that is unbounded in size can certainly be bounded in a lot of other respects, and there is no particularly persuasive reason to think that “every possible combination of atoms is repeated an infinite number of times.”
After all, the set of all integers Z is unbounded and infinite, but this does not imply that every real number occurs inside of it.
However, flat/negative curvature plus the size of the universe being infinite and the cosmological principle, which basically states that the universe lacks a preferred direction and location would imply the conclusion, as it would mean our collection of atoms isn’t special at all, and this goes for any finite portion of the universe.
As a basic counterexample, just consider a fully empty infinite universe. It is in equilibrium (and does not violate any known laws of physics), it has an infinite size, and it adheres to the cosmological principle (because every single region is just as empty as any other region). And yet, it quite obviously does not contain every possible configuration of atoms that the laws of physics would allow...[1]
Or a universe that just has copies of the same non-empty local structure, repeated in an evenly-spaced grid. From the perspective of any of the local structures, the universe looks the same in every direction. But the collection of possible states is confined to be finite by the repeated tiling pattern.
As a basic counterexample, just consider a fully empty infinite universe. It is in equilibrium (and does not violate any known laws of physics)
Your premise violates quantum mechanic, actually. Such an universe’s amplitude distribution is delta function (fully empty with probability 1, any other state with probability 0), which does not have second derivative so its future evolution is undefined.
Actually, collections of atoms (let’s call them structures) can be special.
For instance, there are structures which tend to produce copies of themselves; with some changes (one sign flip), one can obtain structure which tends to destroy its instances. They have approximately same complexity so their rate of randomly arising is equal; however, over time count of former structures increases while count of latter decreases. So we shouldn’t expect all atom collections to appear with equal probability even in full universe.
After all, the set of all integers Z is unbounded and infinite, but this does not imply that every real number occurs inside of it.
This is because Z: 1. Isn’t random
2. is uncountably smaller than R
Saying every finite combination of atoms exists an infinite number of times in an unbounded universe is more like saying every finite sequence of digits exists an infinite number of times in the digits of pi.
This is imprecise. It is more useful to say that it happens because we literally have Z≠R.
Indeed, randomness and countability have little to do with this situation. Consider S=(R∖Q)∪B, where B is a random set of nonzero rational numbers such that P(q∈B)=12,∀q∈Q∗. Then S is a random set (i.e., a set-valued random variable) that is not uncountably smaller than R (the difference between R and S is included in the countable set Q), and yet we know for sure that not all real numbers are in S (because, for example, 0 cannot be an element of S).
Saying every finite combination of atoms exists an infinite number of times in an unbounded universe is more like saying every finite sequence of digits exists an infinite number of times in the digits of pi.
Note that the last property (aka, the idea that pi is normal) is not something that has been proven. So if you are trying to use the notion that it has to be true in an analogy with the idea that “every finite combination of atoms exists an infinite number of times in an unbounded universe” must also be true, this would not be a sound argument.
In light of this particular example, I also don’t really understand why you focused on randomness in your previous comment. After all, pi is not a “random” number under the most natural meaning of that term; it and its digits are fully deterministic.
Indeed, randomness and countability have little to do with this situation.
No, Z literally cannot contain R because R>Z. On the other had, we know that the universe is made up of atoms and that many finite arrangements of atoms can be found in the universe.
Note that the last property (aka, the idea that pi is normal) is not something that has been proven. So if you are trying to use the notion that it has to be true in an analogy with the idea that “every finite combination of atoms exists an infinite number of times in an unbounded universe” must also be true, this would not be a sound argument.
You maybe confusing “sound” with “proven”. Most mathematicians believe pi is normal, so assuming this is a perfectly reasonably assumption to make when reasoning even if the conclusions of that reasoning won’t have the rigor of mathematical proof.
I don’t see what this has do to with randomness or countability? You are the one who brought those two notions up, and that part my response only meant to deal with them.
You maybe confusing “sound” with “proven”.
No, I am using “sound” in the standard philosophical sense as meaning an argument that is both valid and has true premises, which we do not know holds here because we do not know that the premise is correct.
Right, but such an argument would not be “sound” from a theoretical logical perspective (according to the definition I mentioned in my previous comment), which is the only point I meant to get across earlier.
As a side note, I do not like seeing these two possible causes of this belief be put in the same bucket. I think they are quite different in an important respect: while it seems MWI would indeed imply this conclusion, negative curvature alone certainly would not. A universe that is unbounded in size can certainly be bounded in a lot of other respects, and there is no particularly persuasive reason to think that “every possible combination of atoms is repeated an infinite number of times.”
After all, the set of all integers Z is unbounded and infinite, but this does not imply that every real number occurs inside of it.
However, flat/negative curvature plus the size of the universe being infinite and the cosmological principle, which basically states that the universe lacks a preferred direction and location would imply the conclusion, as it would mean our collection of atoms isn’t special at all, and this goes for any finite portion of the universe.
No, I don’t think that alone would do it, either.
As a basic counterexample, just consider a fully empty infinite universe. It is in equilibrium (and does not violate any known laws of physics), it has an infinite size, and it adheres to the cosmological principle (because every single region is just as empty as any other region). And yet, it quite obviously does not contain every possible configuration of atoms that the laws of physics would allow...[1]
Or a universe that just has copies of the same non-empty local structure, repeated in an evenly-spaced grid. From the perspective of any of the local structures, the universe looks the same in every direction. But the collection of possible states is confined to be finite by the repeated tiling pattern.
Unless we use a definition of “possible” that just collapses into tautology due to macro-scale determinism...
Your premise violates quantum mechanic, actually. Such an universe’s amplitude distribution is delta function (fully empty with probability 1, any other state with probability 0), which does not have second derivative so its future evolution is undefined.
Ah, oops.
Actually, collections of atoms (let’s call them structures) can be special.
For instance, there are structures which tend to produce copies of themselves; with some changes (one sign flip), one can obtain structure which tends to destroy its instances. They have approximately same complexity so their rate of randomly arising is equal; however, over time count of former structures increases while count of latter decreases. So we shouldn’t expect all atom collections to appear with equal probability even in full universe.
This is because Z:
1. Isn’t random
2. is uncountably smaller than R
Saying every finite combination of atoms exists an infinite number of times in an unbounded universe is more like saying every finite sequence of digits exists an infinite number of times in the digits of pi.
This is imprecise. It is more useful to say that it happens because we literally have Z≠R.
Indeed, randomness and countability have little to do with this situation. Consider S=(R∖Q)∪B, where B is a random set of nonzero rational numbers such that P(q∈B)=12,∀q∈Q∗. Then S is a random set (i.e., a set-valued random variable) that is not uncountably smaller than R (the difference between R and S is included in the countable set Q), and yet we know for sure that not all real numbers are in S (because, for example, 0 cannot be an element of S).
Note that the last property (aka, the idea that pi is normal) is not something that has been proven. So if you are trying to use the notion that it has to be true in an analogy with the idea that “every finite combination of atoms exists an infinite number of times in an unbounded universe” must also be true, this would not be a sound argument.
In light of this particular example, I also don’t really understand why you focused on randomness in your previous comment. After all, pi is not a “random” number under the most natural meaning of that term; it and its digits are fully deterministic.
No, Z literally cannot contain R because R>Z. On the other had, we know that the universe is made up of atoms and that many finite arrangements of atoms can be found in the universe.
You maybe confusing “sound” with “proven”. Most mathematicians believe pi is normal, so assuming this is a perfectly reasonably assumption to make when reasoning even if the conclusions of that reasoning won’t have the rigor of mathematical proof.
I don’t see what this has do to with randomness or countability? You are the one who brought those two notions up, and that part my response only meant to deal with them.
No, I am using “sound” in the standard philosophical sense as meaning an argument that is both valid and has true premises, which we do not know holds here because we do not know that the premise is correct.
In philosophy it is perfectly normal to use arguments that are merely “likely” to be true (as opposed to mathematically proven)
Right, but such an argument would not be “sound” from a theoretical logical perspective (according to the definition I mentioned in my previous comment), which is the only point I meant to get across earlier.