# Could the Universe have a finite number of states?

I think most people would say the Universe is infinite, and the number of ways of arranging infinite particles in infinite space is infinite.

But we know the "visible" Universe is finite, there is a cosmic horizon. And at some point most of the matter in the Universe will cross this horizon and the information would be smeared on this horizon.

Conceivably the possible states of the Universe at all times might just be finite. And you could number them $$n = 1...$$ a very large number. And our particular universe at this particular time would just be one of the numbers. And we might have a `wave function of the Universe' just $$\psi_n$$.

In mathematical terms $$n$$ might label some huge group or algebra.

In current physics, are the total number of states of the Universe (including states at any "time") considered to be infinite or finite?

For a Universe ending in a big crunch I would say the states would have to be finite. But what about ours?

• If you consider that a particular observer is a computation carried out by a suitable system (for us this is our brain), and these computations are of a finite nature then inside a system with an infinite number of degrees of freedom (not counting degrees of freedom that are not dynamic) there would typically be an infinite number of copies of you. So, this leads to a severe Boltzmann brain problem. Apr 21, 2019 at 3:53
• Isn't there an infinity of possible states even for a very finite universe, e.g one containing a single hydrogen atom, which can take any real for its principle quantum number?
– g s
Jun 17, 2021 at 22:57
• Any positive integer I mean
– g s
Jun 17, 2021 at 23:03

I think the best answer is that science does not know enough to answer this question.

Most of physics is done using real numbers with an infinite number of states. However, we do see integers (and thus our best chance at a finite number of states) in quantum mechanics. In particular, the standard model uses them heavily. They appear because the only stable states for some systems appear with integer numbers.

However, when we start talking about all of the universe for all time, it gets murky. If there exists one real numbered variable which is not selected from the integers, then the universe would have to be uncountably infinite.

The best argument I have heard for there being a finite number of states is Plank time and Plank length. These are astronomically short times and lengths, far beyond anything we can measure. However, we do believe that our models do not describe anything shorter or faster than that. So if it turns out there's nothing new to model, then that would suggest that these Plank times and lengths discretize everything. But when in science has there not been something new to model?

And, of course, you mention the big crunch. Other models do make it harder to tell what the answer should be. If you have a slow cold death, the number of states may balloon, but are they infinite? That's a pretty demanding bar.

The difference between an countably infinite number of states and a very large number is very hard to discern. Indeed, it would take great hubris to feel that we can meaningfully distinguish between Graham's Number and infinity. Yes, mathematics does indeed have that huberis, but science has to deal with things differently. (Graham's number is so big that you can't write it down, 1 digit per atom in the universe. In fact, if you try to write down the number of digits, the resulting number is still larger than the number of atoms in the universe... by an amount so large that its hard to convince yourself this log log process even did anything to the number).

All that being said, if I ask some of my friends in other circles, they might point out that there is only one state the universe can be in: the Is. What is is, and nothing else can be. I give that one for some perspective. The question of "number of states of the universe" does indeed imply that there could exist other universes in different states. But that goes metaphysical very quickly. It just seemed like some good food for thought.

One way this question may be answered is through the finding of a contradiction in the logical consequences of Arithmetic as we expand our mathematical knowledge through theorems and proofs. If I understand correctly, Kurt Godel showed that any attempt to prove the consistency of mathematics by mathematically modelling its logic would necessarily contain an inconsistency within itself, forcing the conclusion that Mathematics is consistent if and only if its consistency cannot be proven. Most mathematicians seem to be willing to accept the consistency of mathematics on faith as, after all, we seem to live in a mathematically defined universe. But it seems to me that even if there is a contradiction, it would only be found in subpatterns of mathematics containing sufficiently large amounts of information, and that such a contradiction would not prevent a limited finite universe operating within a protected smaller subpattern - a finite automaton - from existing, such as the universe within which we find ourselves. It will be interesting to see what happens if (when?) our computer-assisted calculations encounter such a contradiction.