I tried to edit my previous while you were posting I guess, but the time had expired so here it is in full. I agree that it is a cardinality comparison, and I don’t think it possible.
If the number of atoms, electrons, quarks, etc. in the universe is countable (That is equivalent in size to the number of integers: cardinality aleph null), but space-time is continuous (That is equivalent in size to the set of real numbers which is “larger” than the set of integers: cardinality aleph one), then the answer would have to be ‘no’ as the number of arrangements of just two particles would have to be at least cardinality aleph one based simply on their (real number) distance from each other and ignoring any other descriptors.
Now it’s possible that the number of particles in an infinite universe is has cardinality aleph one, or that space time is discrete (at the Planck length or something) and so its cadinality is aleph null, but I still don’t think the answer is yes. It would seem to me (but it’s been too long since set theory) that the set of all arrangements of an infinite number of partices would be at least one higher cardinality than the cardinality of the set itself.
If I’m correct about this assumption, then for all arrangements of particles to be someplace, the cardinality of the set of particles would have to be at least one larger than the cardinality of space-time. But if this is true you cannot put the set of particles into a one-to-one correspandance with the set of points in space-time. You have “too many” particles. So I don’t see how you can possibly fit a higher cardinality of particles into a lower cardinality space-time as each has to be someplace, and (presumably) two can’t be at the same place.
I agree that this is the case. However, this is considering the universe as a whole. Even an infinite universe can only contain itself. It cannot contain all possible infinite universes.
However, the question I’m interested in answering is not about the universe as a whole but a specific subset of the universe. Let’s pick a sphere 10,000 miles in diameter since that’s a volume large enough to hold the Earth.
Imagine the entire universe divided up into theses spheres. If space/time is quantized then there are a finite number of configurations of matter that can exist within one of these spheres. The laws of physics make some of these configurations likely, and some very, very, very unlikely.
Must every configuration of matter that can be contained within a 10,000 mile sphere exist somewhere in the universe?
When we’re dealing with an uncountable probability space, saying that an event happens with probability 1 (we also say it happens “almost surely”) doesn’t mean that it must happen. Similarly, saying that an event happens with probability 0 does not mean that it cannot happen.
For example, pick a number at random from the set of reals between 0 and 1. What is the probability of you picking exactly 1/2? The probability is 0. But it doesn’t mean that this cannot happen. Similarly, the probability of you picking something else than 1/2 is 1, but as we can see, this isn’t what will necessarily happen.
This is related to what we know in popular culture as the infinite monkey theorem. If you have a monkey typing at random on a typewriter, for an infinite amount of time, any given finite string of text will be found somewhere within the output of your typing monkey with probability 1. However, it doesn’t mean that it must be found in the monkey’s output. After all, the monkey could very well only be typing the letter ‘a’. The probability of this happening is 0, of course, but it is a valid result. This is because the space of all infinite sequences of letters is uncountable.
The current state of the art in cosmology is that we know of no experimental bounds on the size of the Universe, and in fact it is unlikely that we will ever be able to determine any such bounds. In other words, we’re quite sure that we don’t know whether the Universe is infinite or not.
I don’t see how this could be possible. Even in an infinite universe, I could construct a list of all of the particles in the universe, just by starting with the ones closest to me and working out. Such a list would constitute a mapping of the particles to the integers, so the particles would be countable.
You’ve argued circularly here. Yes if you can make a list of the particles, then by definition they have cardinality aleph null. But you’ve simply assumed you can make such a list. I happen to agree with you that you can make such a list in our universe so that the cardinality is aleph null. In fact I also believe the list is finite. But neither your nor my belief makes it so.
If the laws of physics still apply, then I imagine the answer is no because there are configurations of energy and matter that are not allowed to exist according to the rules.
Although some answers are already over my head (not that tall an order, really), here goes my venture of an answer. In an infinite number of universes, everything that can happen is likely to happen (not certain, as previously explained). The impossible, though, will not happen.
Why would there be an Earth where bowling balls float an inch over the floor? That would be in violation of the laws of physics. It cannot happen.
Your idea of the volume and the arrangement of particles inside. You are thinking of it as akin to a computer screen. Any combination of pixels is possible. Think of it as a section of Conway’s Game of Life. Some combinations are not naturally reachable (Garden of Eden configurations). No matter how many times you try, you will not see those configurations appears spontaneously.
Added: I see that RaftPeople beat me to the short version of it. Nice
Assuming an infinite number of universes, why can’t the laws of physics vary from universe to universe? We only know how physics works in our universe, and the “laws” are really just descriptions.
Not circular. The first particle on my list is the one that’s closest to me. The second particle is the one that’s next-closest, and so on. Show me any particle in the Universe, and I can figure out its number: I just take a sphere centered on me, and with a radius equal to the distance to that particle, and count all of the other particles inside of the sphere.
oh well, if we are changing laws of physics (which are, of course, shorthand for “the realities that we describe with our science”), then anything goes, I guess. I just assume that if we are bothering to keep the number of particles constant, then the laws would also hold.
This makes sense to me, if we assume that only finitely many particles can fit within any given sphere. Which sounds like a reasonable assumption, but I don’t know enough about particle physics to know whether it is one or not.
But is it impossible, or just very, very unlikely? There’s a great deal of molecular motion in a bowling ball at room temperature. Presumably if enough molecules randomly headed up at the same time the bowling ball would levitate.
It’s like a varient of Maxwell’s Demon. Entropy is a statistical effect. Take a small enough sample, or go far enough out on the tail of the bell curve and you may enter a domain where entropy locally decreases rather than increases.
Aren’t you assuming that space-time is reasonably flat everywhere to be able to do this? Could space-time be sufficiently curved (if only locally in some places such as near a rotating black hole) so that it is not a meaningful statement to say that there is a single distance between point A and point B?
I think that when people talk about an infinite universe they’re not talking about our local universe but the "Multiverse "of universes.
Surely infinity DOES exist in our universe ,I can see no reason that we yet understand why you cant continue doubling Numerals forever.
And the Ancient Greeks philosopher (cant remember his name sorry) who posed the question
"Why can you fire an arrow from A to B when theoretically it should be impossible ?
As to cover the full distance the arrow has to cross half the full distance first,
and to cross half the half distance it has to …and so on Ad infinitum.
I didn’t put it very well but basically its mathmatics again in that however small the measurment it can always be halved .
I’ve wondered if,sometimes ,subatomic particles will continue to be discovered smaller and smaller forever!
Trying not to belabour (Brit spelling)the point but am probably doing so its likewise with the infinite possibility of variations/choice as we navigate through time.
There also is no reason whatsoever to believe that the laws of Physics as we know them ,must be applicable through out all the universes (and some have said throughout our local universe but I’m skeptical about that one myself)
Close to us in comprehension there could be universes where atoms are small hard balls,where magic,alchemy and religion themselves are Laws of nature but further afield we probably just cant get our imaginations round them.
The sheer illogicality and variety in our local U. suggests that the larger U.is composed of infinite variety rather then infinite repitition ,and a finite U.while easier for human minds to grasp rather then the alternative doesn’t work for me .
Since we have departed from the realms of logic, this is how I think miracles happen. Jesus walked on water because tau-pico-muonettes aligned themselves in just the right way for him not to sink. Or a rogue volcano in the Indian Ocean rerouted a turtle migration and Jesus walked on their backs. Or something else equally improbable.
Now for a planet where bowling balls all float one inch above the ground, that goes beyond lucky coincidences, I would think. I know I am complaining about the unlikely shape of a brick in the Harry Potter world, but still. Even my fancy has some limits.
Yes and no. Certainly there can be multiple paths of different lengths, but you can have that even in a flat universe. If there are multiple paths, the convention is to define the “distance” as the length of the shortest path, or the limit of lengths of progressively shorter paths (if the “shortest path” doesn’t exist). The limit, at least, will always exist, no matter how crazy the space is.
For the purposes of this particular line of inquiry, I’m NOT talking about the multiverse. I’m talking only about our own universe … which may very well be infinite.
I’m sorry, but I really don’t see why this is “beyond the realms of logic”. Mathematics deals with various orders of infinity all the time, as well as vanishingly small probabilities. (For example, I’ve read articles postulating the long-term fate of the universe assuming that protons decay. The half life of a proton is estimated by some theories to be 10^35 years – making its decay a very low probability event indeed.)
Random molecular motion resulting in levitation is definitely a very low-probability event. But as far as I know there’s nothing in the laws of physics that makes it impossible.
To answer the question of the title of the OP, the answer is no.
In an infinite universe, assuming a finite number of possible configurations of any given finite volume, there must be repeated configurations, but there is nothing forcing every possible configuration to be actualized.
Say there are three possible configurations, A, B, and C. The infinite universe might well look like this:
…ABBAABBBAABAAABBBAAABABAABBBAABAAABA…
and so on, with no actual instances of configuration C.
I shudder to think of a universe filled with clones of ABBA, but nevertheless, we learn it is possible from SCIENCE!.
