# Can we assume that there are infinitely many possible states of the world at time t + 1?

I wasn't exactly sure where to post this question, so I decided that it's sufficiently philosophical in nature to warrant its being posted here. I may end up posting it on Mathematics SE as well.

I've been trying to model time and the space of all possible moments as something a computer scientist would consider a tree. That is, the tree has a root node, which signifies the instantiation of time, and the root node has children, all of which are possible proceeding instances. Further, each of those instances has infinitely many potential proceeding instances. This obviously yields a tree whose nodes all have infinitely many children.

Now I'm a bit worried. Here are some of the implications of such a model:

• Each node has a history.

• Sibling nodes have the same history.

• Thus, one particular history can lead to infinitely many proceeding instances.

How could it possibly be the case that the same history could lead to different states of the world? One possible conclusion is that the assumption that "there are infinitely many possible states of the world one instant from now" is naive and ultimately incorrect, in which case "there are finitely many possible states of the world one instant from now" would be correct. In THAT case, there is either one instant (which would imply that all of time is on a very particular, unchanging and predetermined course) or some finite number of instances greater than 1.

My question is the one posed in the title–is there something wrong with the assumption of infinitely many possible states of the world? If anything is unclear, make note of it and I will try to clarify.

• I understand how the topic you're thinking about has philosophical aspects but the question you're asking is really about mathematics. I think at the heart of it you're just asking about what a phase space is. If a system has an infinite amount of degrees of freedom then it has an infinite amount of possible configurations it can be in. You're using the term world but I assume you mean universe, so does the universe have an infinite amount of degrees of freedom when it's understood as a physical system? Yes, I believe that is clearly the case. – Not_Here Feb 17 '18 at 6:12
• I have a dissenting view to the first comment. Even in the model being described, the laws of physics would have to hold consistently, putting a limit to the number of possible outcomes from an existing system. A bowl of petunias and a sperm whale can't just appear out of thin air, for instance. That very limit means that the word infinite cannot apply here. – Tim B II Feb 17 '18 at 10:52
• @TimBII A bowl of petunias and a sperm whale are objects that are made out of fundamental particles, fundamental particles are excitations in the underlying quantum fields, quantum fields undergo random fluctuations (a fact that has been empirically verified). We're talking about the entire degrees of freedom of a system, not just some chosen macroscopic parameters. If the universe is infinite, as in it has an infinite volume, and the quantum fields permeate the entire volume, then yes it is completely within the bounds of physical laws for there to be an infinite amount of states. – Not_Here Feb 18 '18 at 1:29
• Given a time t the universe is in state A, at time t+1 you can cycle through an infinite amount of possible states B just by iterating through quantum tunneling and quantum fluctuations. If position (x,y,z) in the QED field is in a vacuum state at time t, then at time t+1 it fluctuates to having an excitation. If (x,y,z) is in an excited state already at time t, then at time t+1 it quantum tunnels to a different energy level. You can iterate through every (x,y,z) doing this, which results in an infinite amount of possible states if the universe is infinitely large and the fields permeate. – Not_Here Feb 18 '18 at 1:34
• @Not_Here - You're assuming an infinite universe here, which is not guaranteed, nor would it ever be observable so there is no way to prove this. Also (and this is the bit that is confusing me), if fundamental particles are really just excitations in the underlying quantum fields, then they are manifestations of the underlying state of the field, and given that laws like relativity have been consistently observed at the classical scale, surely that limits the number of manifest states in the underlying field. That is to say, only certain (non-random) fluctuations can exist. – Tim B II Feb 18 '18 at 6:04

Consider that many of the possibilities 'amount to to the same'. It is likr degenerate energy levels in a quantum system. Or states which increase entropy in a similar way - say you have a container half vacuum half gas and remove the divider; there are lots of states if you think of every atom as numbered and distinguishable, but are they? This thinking leads to Bose-Einstein statistics.

Wheeler, who was the supervisor to Everett's Many Worlds interpretation of quauntum mechanics PhD thesis, pointed out the Achilles heel. Shannon entropy shows the link between information, energy, and available states. So if there are infinitely many branches, where is all the energy for the information in them coming from? There must be a limiting factor, perhaps at the particle scale (things only branch like this at the quantum scale, not above), or another interpretation entirely

Your question relates heavily to physics. I suggest you give Stephen Hawkings' "A Brief History of Time" a read. Meanwhile, though, to explore your question, consider this: nothing can move faster than the speed of light. Therefore, whatever distance of time there is between t and t + 1, anything can move in any direction only as quickly as just below (or at) the speed of light in the worst case. As a result of that, every possible instance of the universe can be collected finitely. (It is also important to note that by the Isolation Principle, no two particles can ever occupy the same space while moving at very nearly the same velocity and direction)

Where you run into a problem is in your assumption that all things with the same history must conclude at the same result. Something I'd highly recommend you take a look at when considering this is the Heisenberg Uncertainty Principle. It essentially states that we can never exactly predict the state of the universe, even if we know exactly how it is or was at one point in time. This has a lot of profound implications, but exploring those is beyond the scope of this question.

Another important thing to keep in mind is that the universe is not infinite. Not at all. The universe is what we describe as finite without boundaries. It consists of a collectively finite amount of matter and energy, and exists within a finite space, but is not technically confined.

• "As a result of that, every possible instance of the universe can be collected finitely." → Where does this claim come from? – Veedrac Feb 19 '18 at 0:32
• @Veedrac Apologies, maybe I did not explain that thoroughly. I am essentially saying that everything has a finite future, because they are confined to the distance at which they can move at the speed of light. As a result, we can finitely collect all of its possible futures, and thus all possible futures of everything else. Determining the likelihood of arriving at each, though, is a different question. – Dallas Crenshaw Feb 19 '18 at 0:35
• That argument would also show there are a finite number of values between 0 and 1. – Veedrac Feb 19 '18 at 0:38
• @Veedrac Not necessarily true, particularly when you consider the Isolation Principle. – Dallas Crenshaw Feb 19 '18 at 0:39
• "Not necessarily true" → So what is your actual justification? – Veedrac Feb 19 '18 at 0:40