# Are there really infinite alternatives? [duplicate]

This question is an exact duplicate of:

The Multiverse theory says that there are infinite alternatives of our universe, many of them are very similar to the one were we live other are completely different. But the number of parallel universes isn't infinite. It' s a very, very, very.... huge number because we have to consider all the possible configuration of the universe (for example the number of the particles at the power of itself multiplied by....) but at all it shouldn't be infinite.

Is this wrong?

## marked as duplicate by AccidentalFourierTransform, John Rennie, Qmechanic♦Dec 1 '16 at 19:08

This question was marked as an exact duplicate of an existing question.

• the multiverse theory is not a scientific theory, it is just sci-fi. – AccidentalFourierTransform Dec 1 '16 at 16:54
• I am very sceptical about theory that claims that the number of possible configuration of the universe is finite. It admits implicitly that space and time are discrete (not continuous) which nobodies knows but seems unlikely. See for example this question. – user130529 Dec 1 '16 at 17:01
• There are two ways in which I could answer your question: One, from a philosophical perspective, the idea of "other universes" is meaningless. Since, by definition, such other universes will not interact with our own in any way at all (if they did, they would in fact have to be considered part of "our" universe), nothing can be said about them. Second, as a physicist, I would say that, again by definition, "other universes" are unobservable. This means I cannot apply physics to them, just as I cannot apply physics to dreams and imaginations. – Pirx Dec 1 '16 at 17:19
• Please can you identify the source of your information about the Multiverse Theory. It does not seem to be mainstream physics. If there is no accepted Multiverse Theory in cosmology, we cannot explain what it says. – sammy gerbil Dec 1 '16 at 17:24
• This is a disaster of a comment section. This is not the place for extended discussion (including sharing questionable opinions about the validity of theories which are, if nothing else, certainly part of the literature). Additional questions certainly do not belong here. – DilithiumMatrix Dec 1 '16 at 17:31

As AccidentalFourierTransform pointed out, the multiverse theory is not a scientific theory, it's science fiction. There is a tangentally related scientific theory called the many worlds interpretation of quantum mechanics, but that is a very different question.

However, despite this, there are a few mathematics lessons to be had here, because infinity is a curious thing. Mathematicians have a very precise set of definitions regarding infinity, and some of them may be counterintuitive.

The first thing to point out is that many of our models of the universe include real numbers. If I say "This ball is 3.005cm away from this pen" I'm doing a measurement, limited by my own accuracy. The models themselves are not limited by my measurement accuracy, suggest that the distance between the ball and the pen could be any real number. This means my simple ball/pen system already has an infinite number of possible configurations! Uncountably infinite, in particular, because there's no way you could count all the possible positions, even if your process of counting took an infinite number of steps.

In quantum mechanics, many of the models involve integers instead of real numbers. If you "define" the universe based around these integers, you will have a countably infinite number of possibilities. This means that you could count all the possibilities, if you were willing to take an infinite number of steps to do it. A bunch of modeling assumptions aside, and ignoring the nuanced difference between countably infinite and uncountably infinite, we still would have an infinite number of multiverses.

But what if we consider just the simplest example. We look at all the particles in the universe, and make a big table. In each alternate universe, each particle is either present or not present. If we did this, we could assign each universe a name, using the letter "P" for present and "n" for not present. Thus, in a 5 particle world, PPnnP would be one universe, and PnPnP would be another.

You seem comfortable with the assumption that there are a finite number of particles in the universe, so we'll run with that. That means there must be a finite number of possible worlds named in this way, right?

Not necessarily.

If we assume that our universe is the "seed" for this multiverse, meaning we have all the particles (PPPPPPPP), then it's pretty clear there's a finite number of worlds out there. However, what if we don't make that assumption. What if the only assumption we make is that "all universes can be named by a finite string of P's and n's?" This clearly shows that each universe has a finite number of particles, because its string of PnnPPnP... must be of a finite length, so the number of P's in that must also be finite. And it also permits the existence of universes which have particles we don't!

It turns out, by choosing my construction carefully I've constructed a formal language consisting of {P,n}*, that is to say "all finite strings containing P and n." Surely since the length of each string is finite, and the alphabet is finite (2 characters, P and n), the number of possible strings naming these universe must be finite, right?

Nope.

It turns out that when you handle that construction formally, you find that the number of finite strings built up from 2 characters is countably infinite! So even in this incredibly limited case where I required every alternate universe be labeled with a finite string, we still find infinities.

Mathematics is weird.

Don't believe me, or want to know more? I typically point people towards VSauce's videos on the matter. His explanations are far cleaner than mine, and I love his visuals:

• How to Count Past Infinity - for an introduction to the weirdness of how mathematicians handle infinities
• I don't have a good video on the "Kleene Star," which is the tool I used in my formal language argument, but the Wikipedia article is pretty good.
• The Banach-Tarski Paradox - VSauce uses the same sort of formal language trick as I use here to show one of the more famous paradoxes in set theory. It's a little harder to understand, but it uses the same principles as I used here.
• The "weirdness" you are talking about here simply comes from the fact that you allow finite strings of arbitrary length. There is nothing weird about this at all. As soon as your restrict your language to strings of a definite length $m$, the number of possibilities becomes finite, and trivially so. – Pirx Dec 1 '16 at 17:32
• @Pirx Oh yes, that is exactly true. But to me, it's kinda wierd that the number of strings whose lengths are all finite, built from a finite alphabet, is countably infinite. The intuition in me is that you can't build an infinite thing from a bunch of finite things (just as the OP felt), and the weirdness is the carefully selected behavior of the kleene star which lets mathematicians do exactly that without the world imploding on them in a filthy pile of inconsistencies! – Cort Ammon Dec 1 '16 at 17:40
• Similar to how the Axiom of Choice is totally reasonable looking, until you add infinite sets, and then it starts generating really odd results that make you question whether you could safely assume the AoC in the first place. – Cort Ammon Dec 1 '16 at 17:42
• +1 for mentioning how mathematicians handle infinities. I I have always thought that physics never had to bother with Cantor, as there is no proof, afaik, that there is/are an infinity of physical "things" to worry about. i hope I am right in my simplistic view of it. – user108787 Dec 1 '16 at 18:21