How to determine cardinality of an infinite set using Aleph numbers?

Question

So I was reading a little bit about cardinal infinities, and I thought it was pretty interesting. However I wanted to know a little bit more about how to use them. For example, how would I determine the cardinality of the set of all real numbers? The one thing I know is that if it is countably infinite, then the cardinality is $\aleph_0$, and if it is uncountably infinite, then the cardinality is $\aleph_n$ ($n>0$). I understand the difference between countable infinity and uncountable infinity, so therefore the difference between between $\aleph_0$ and every other $\aleph$ number. But the problem is that I don't know the difference between any of the uncountable ones. What is the difference between $\aleph_2$ and $\aleph_3$? And how can I determine which $\aleph$ number is the cardinality given any uncountably infinite set?

Answer 1

Given a finite set. Can you determine its cardinality?

No, not really. If I tell you that it's not empty, can you say much more? You can say its cardinality is nonzero, but that's about that. It could be anything else.

Uncountable sets are just sets which are not countable. Their cardinality could be anything. Assuming the axiom of choice, that cardinality is some $\aleph_\alpha$ for $\alpha>0$, but that's pretty much all you can say without further constraints.

But even then, the standard axioms of set theory are quite weak when it comes to deciding the actual cardinality of particular sets of interests. The reason is that the move from $\aleph_0$ to $\aleph_1$, or from $\aleph_2$ to $\aleph_3$, is a move which happens on the ordinal numbers. But the sets we are usually interested in are not represented as sets of ordinals to begin with (in most cases, anyway).

Instead, we usually find ourselves interested in power sets, and power sets of power sets, and the standard axioms of modern set theory are not sufficient to established what is the exact value of $2^{\aleph_0}$, or $2^{2^{\aleph_0}}$. We can give some constraints, but not much more than that; at least without additional assumptions. This sounds strange, but at the same time the axioms of a field are insufficient to determine whether or not $2$ has a single $7$th root, or more, or none. So being unable to determine something like that is not terribly unsettling throughout mathematics (even if sometimes we feel that it is).

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What is the difference between the $\aleph$ numbers? (Which can be indexed by more than just the natural numbers)

For that you need to slightly understand what are ordinals. $\aleph_0$ is the size of the smallest infinite ordinal; $\aleph_1$ is the number of ordinal of size $\leq\aleph_0$; and $\aleph_2$ is the cardinality of the set of ordinals of size $\leq\aleph_1$; and so on. The difference is "essentially formal", and mostly intangible. We can only say that $\aleph_3$ is much larger than $\aleph_2$, but not much more than that, for some very excellent reasons which are difficult to explain with additional background in logic and set theory.

Could someone explain aleph numbers? might help you understand slightly better the $\aleph$ numbers.