Cardinality of the Universes Set No expert by any means, but sometimes, in different contexts the term multiverse used. In quantum mechanics, some say that it is possible that there are actually many universes where all the possible states are manifested, cf. the many-worlds interpretation. I wondered what is the cardinality of the set of all possible universes? Does the answer to this question has or can have any theoretical consequences?
 A: One easy way of working out the cardinality of the "multiverse" (which does not have a unique, concrete model behind it) would be to think of how the "universes" within the multiverse are labeled. Suppose that your multiverse is based on different choices for some physical parameter $\Lambda$, and each universe would have its own setting for $\Lambda$. Then, the cardinality of the multiverse would be determined by the cardinality of the set of possible choices for $\Lambda$. Therefore, if you could only have $\Lambda \in \{1,2,7\}$, then there are clearly three possible universes; if you can only have $\Lambda \in \{1,2,3,\dots\}$, then there are countably infinite universes, etc.
Physical consequences of the cardinality of this set are of course determined by whether or not the universes interact. In both the many-worlds interpretation of QM and the string landscape as a multiverse pictures, there are no interactions between the universes so in those cases, there are no physical consequences.
A: The many worlds interpretation assigns reality to every possible outcome of a measurement of the quantum state. So consider for example a Stern-Gerlack experiment on a single electron, measuring in some fixed axis, you get two outcomes to this experiment, so in the many worlds interpretation a complete description of reality would include two worlds. This example can be more concretely understood in terms of measurement operators and eigenstates of those measurement operators. Here our Stern-Gerlack experiment is a Pauli operator, and the Pauli operator has two eigenstates. So we can say that a measurement creates as many worlds as it has eigenstates. Since each measurement compounds the number of worlds, the cardinality of the multiverse must be at least as large as the cardinality of the largest possible set of eigenstates belonging to one possible measurement. I suppose the measurement with the largest number of eigenstates would be a measurement of a system with an infinite dimensional Hilbert space, such as measurement of position, which would have an infinite number of eigenstates. I can't claim to know what exactly the cardinality of that infinity must be, but if I were asked to guess I would say it is uncountably infinite.
A: Countable? No way.
It is certainly at least $\mathfrak{c}=2^{\aleph_0}$ (the cardinality of the continuum).
The number of possible states of the simplest system imaginable in just one universe is at least $\mathfrak{c}$, as spatial coordinates are real numbers.
There are many fanciful (and therefore fashionable) notions that space and/or time might be discrete, but none with any supporting evidence.
I imagine that the cardinality of the multiverse is in fact just Aleph, just as the cardinality of $n$-dimensional reals ($\mathbb{R}^n$) is still just $\mathfrak{c}$ for any $n$.
A: The answer is "it depends". If you're talking about the set of distinguishable universes, then to calculate the number of universes that could be produced you would do something like the calculations of entropy in "Universal upper bound on the entropy-to-energy ratio for bounded systems":
http://old.phys.huji.ac.il/~bekenste/PRD23-287-1981.pdf
If you look at some particular member of the set of distinguishable universes, it has a real valued number that represents the probability of getting that result.
So the cardinality of the set of universes is either a finite number or the cardinality of the real numbers.
A: Here is a simplistic mathematical possibility.
If there are $T$ timesteps and $K$ possible choices that can be made at each point, then there are $K^T$ different multiverses that could exist.
Taken to the limit, $T$ should equal the cardinality of the continuum $\mathfrak c$. Collapse of a quantum wavefunction could contain eigenvalues that also count over the continuum $\mathfrak c$, for example for the position & momentum operators (and not energy operator, which could be countably many). This gives a naive answer: there are at most $\mathfrak{c}^{\mathfrak c}$ multiverses. There are probably more educated guesses / calculations, for example considering the lack of global time coordinate in the universe.
