# Is the number of universes finite, countably infinite or uncountably infinite (and what size of uncountable if so)?

Assuming that the alternative universe theory is correct, how many alternate universes are there?
From my understanding an alternate universe "pops up" whenever a particle goes from being in an undetermined probabilistic state into an actual state. For each of the other states that it could have been observed to be in there is a universe where that observation is made. From this view point, I would think there would be a finite number of universes, infinitely countable or unaccountably many (size of the continuum) depending on whether or not there infinitely many positions an electron can be in at any given time (for example). The size of the continuum if the universe is continuous (infinitely many points between any two points) , countably infinite if the universe is discrete and infinite. There would be only finitely many if the universe is discrete and finite. If any of these alternate universes were to be continuous then there would be unaccountably many universes.

If any one of the universal constants could be changed by an infinitesimal amount and still have a valid universe and for each set of universal constants there was a unique universe then clearly there would once again be an uncountable amount of universes. But what size of infinity?

What if every internally consistent set of laws is considered to be it's own little universe, how big of an infinite then? It seems to me that if you say it has such and such cardinality I could construct a larger set of possible laws that are larger.

Is there an accepted theory on what defines what an alternate universe can be?

• In the the last case (is it the tegemark's ultimate kind of multiverse right?) i think that we should look for the class of all the mathematical structures that describe a universe ... but then the cardinality of that collection of "universe" structures would depend on the background foundation theory chosen i guess.. Really interesting question ... i can't wait for the answer. Commented May 30, 2014 at 22:27
• There's no reason to believe that Zermelo-Fraenkel set theory is valid in the physical universe. Commented May 30, 2014 at 22:43
• (continued) What I mean is that it's ZF that allows us to define infinite sets; lets us prove that infinite sets exist (we simply declare an axiom to that effect); and allows us to classify sets as countable or uncountable; and gives us the wild world of transfinite ordinals and cardinals. There is no reason to believe any of this applies to the real world; so the question of the cardinality of universes in the multiverse may well not be approachable in terms of ZF at all. Concepts such as countability may not even make sense in the physical world. Commented May 30, 2014 at 23:45
• Maybe in THIS universe your right, what about the other ones? Commented May 31, 2014 at 2:38
• Having personally counted all the universes, I can assure you that it is a very large number - uncountably large. Commented Mar 7, 2022 at 16:42

It looks like you're referring to the Everetts many worlds interpretation of Quantum Mechanics. This is a solution of the wavefunction collapse, where one can say a measurement has been made. This possibly requires a little elaboration.

First, measurement is not just what a physicist does when he measures the momentum of an electon or an atom, but what every system does when it reacts to another: One could say here, that the first system 'measures' a property of another system and reacts accordingly; but it is also true that the second system measures the first system and also reacts accordingly.

Secondly, and putting it simply, one could say - at least intuitively - that the wave-function represents the possibilities which hold until a measurement occurs, and then the wave-function 'collapses'. One ought to note that the wave function evolution is deterministic until collapse, and this collapse is non-deterministic; and so a certain value is chosen at random for the measured property.

Classically, since Newton, physicists have expected that the universe to be deterministic and real and this held upto the discovery of General Relativity by Einstein. Quantum Mechanic broke this paradigm it seemed irretrievably and the locus of the problem seemed to be the non-deterministic (ie random) evolution of the wave function collapse and also the interpretation of the wave function as possibitities.

It was Everetts aim to retrieve this deterministic & real character for the then new Quantum Mechanics. He posited then every collapse engenders a new universe. Thus a possibility is no longer a possibility but another dimension of reality. This seems a rather high cost for determinism and realism.

As an interpretation it is intriguing but esoteric, and at least physically one requires something more; does this picture of reality provide us e*explanatory power* - Everett attempted to provide one by deriving the basic Borns Rule from it. There is no consensus as to whether this has been done.

Its worth pointing out, given the eye-catching, media and science-fiction friendly parallel world vista of Everetts that a different and much less well known intepretation, Bohmian Mechanics also retrieves realism and determinism by allowing non-locality - that is faster-than-light signalling.

Now, given the tremendous success of the atomic hypothesis in modern physics, and this covers not just the idea of the classical atom, but also quanta (for are they not discrete?) it seemed only natural to think that perhaps even the very structure of space & time is atomic (there are various research programmes that look at this - spin foam and causal nets), and one expects this structure to appear at the Planck scale.

Then given that there has only a finite time has passed since the creation of the universe (the Big Bang), it appears only a finite number of universes are possible, though their number is increasing exponentially.

Its intriguing to consider what kinds of conditions might want to consider that allows either a countably infinite number of universe, or simply (!) uncountable. Personally, my intuition would be that at least one of the basic categories of physical materialism - matter/energy, space-time and gauge-forces are infinitely divisible.

But one also should consider that a rule-of-thumb operates in Physics, which is that infinities are to be avoided: one does not have infinite energies, nor an infinite past, and nor an infinite amount of matter, and nor even an infinite expanse of space. On that basis one might want to rule out an infinite number (of whatever cardinality) of worlds.

• It seems to me that under this theory there are either finitely many universes or unaccountably many. Imagine that a new Universe is created every time an electrons' spin "choses" up or down. Now either this happens finitely many times, say n, in which case there are 2^n universes. If somehow this could happen any an infinite number of times then this would be 2^(set up whole numbers) which is our first uncountable set,R. Commented May 31, 2014 at 3:45
• Sure; but given what we know about physics - its atomic nature (even space-time is, speculatively, considered to be atomic) - it appears more likely that n is a tremendously large but still finite number. As I explained its natural to consider what the conditions would be for n to be infinite. Commented May 31, 2014 at 9:16
• And my intuition would be that least one of the fundamental categories of physics would have to be infinitely divisible ie spacetime, matter/energy or forces. Commented May 31, 2014 at 20:33
• @Michael Higgins If there are uncountably many of anything physical, then the Continuum Hypothesis becomes a fact of the universe, in principle decidable by experiment. I find that impossible to believe. Commented Jun 2, 2014 at 18:21
• @user4894: I don't think this is right: The continuum hypothesis is a purely mathematical statement. The finitness, countability or uncountability of allegedly parallel universe makes no difference. Commented Jun 3, 2014 at 1:42

It depends, of course, on what you mean by a universe, but if what you mean is, for example, a Lorentzian manifold with a connection that satisfies the Einstein equations, then such things form a proper class, not a set, so it's meaningless to talk about their cardinality.

In eternal inflation, the number of hypothetical universes is finite if you believe the multiverse had a beginning.

You can't start from nothing and add up to an infinity. Nevertheless, many multiverse theorists imagine the number of worlds in their theories to be infinite. Alan Guth, for example. They rarely address the countability question, but it seems to me that they would have to be countably infinite because you could write them down on an infinitely long list.

It's a different question with respect to laws. Here the free parameters of our world, which are pure numbers, have irrational values. The set of irrational numbers is uncountably infinite. Therefore, there must be more laws than worlds.

Your question mixes up two different sense of the idea of alternative universes. One is the multiverse as described by quantum mechanics. The other is universes with different sets of laws of physics. With respect to the latter it's difficult to say much because the theory is not well developed.

In quantum mechanics, different instances of the same system can undergo interference, e.g.- a single photon in a double slit experiment. If you look at the interference pattern in such an experiment and you look at a given point there is no single fact of the matter about what slit the photon came through since instances of photons from each slit interacted to give the final result. So the interference pattern does not contain information about what slit a photon came through. By contrast, if I type the letter z then you can tell that I have typed that letter, so information about what letter I typed is copied from my computer to you. A universe is a structure within the multiverse in which information is copied. So there is a universe in which I typed the letter z because there are lots of copies of the information that I typed that letter. There is not a single universe in the interference experiment wrt which slit the photon went through because that information is not copied.

The information that can be copied in that way is discrete, so the number of universes is discrete. See section 2 of

Continuous quantities may be relevant to the probabilities of a given set of universes, but the set itself is discrete.