# “Real” cardinality, say, $\aleph_\pi$?

Is there any meaningful definition to afford for $\aleph_r$ (as in cardinality) where $r\in\mathbb{R}^+$? $r\in\mathbb{C}$? What about $\aleph_{\aleph_0}$? Can we iterate this? $\aleph_{\aleph_{\aleph_{\cdots}}}$

I may be throwing in bunch of rather naive/basic questions, for I haven’t learnt much about infinite cardinalities. If I am referring to bunch of stuff abundantly dealt in established areas, please kindly point out.

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The $\aleph$ numbers are well-ordered. This means that every non-empty set has a minimal element. Furthermore, they are linearly ordered.

This means that any indexing imposed on them should at least have these two properties. The real numbers are not well-ordered (consider the subset $(0,1)$, or even $\mathbb R$ itself) and the complex numbers are not even ordered in any natural sense.

The idea behind having a well-ordering is to say what is the next cardinality. Given a set, we can easily tell what is the least $\aleph$ which is larger. In the natural numbers (and their generalization, the ordinals) we have a successor function which does that, so it is a good ground to use when indexing cardinalities. We don’t have a nice successor function for the real numbers, or for any dense ordering for a matter of fact.

It is possible to have $\aleph_{\aleph_0}$, but there is a minor problem here. $\aleph_0$ is the notation discussing size, whereas $\aleph_\alpha$ is the cardinal that the cardinals below it have order type $\alpha$. So we write $\aleph_\omega$, where $\omega$ is the least infinite ordinal. This is a limit cardinal, which means that it is not a successor of any cardinal — but there are smaller $\aleph$’s nonetheless.

This of course can be reiterated, but we need to use the ordinal form, rather the cardinal form. Namely, every $\aleph$ number is also an ordinal. $\aleph_\alpha$ is the actually the ordinal $\omega_\alpha$, where these ordinals are defined recursively as ordinals which are $(1)$ infinite; and $(2)$ do not have an injection into any smaller ordinal. The least is $\aleph_0$ and it is the cardinality of the natural numbers, which is the ordinal $\omega$.

So if we wish to iterate, $\aleph_0\to\aleph_{\aleph_0}\to\aleph_{\aleph_{\aleph_0}}\to\dots$ we actually need to do it as following: $$\aleph_0\to\aleph_{\omega}\to\aleph_{\omega_\omega}\to\ldots$$

That been said, without the axiom of choice it is consistent to have sets whose cardinality is not an $\aleph$ number, namely sets which cannot be well-ordered. It is consistent that there is a collection of sets which is ordered (by inclusion) like the real numbers, and no two sets have the same cardinality (there is no bijection between two distinct sets).