Set whose cardinality is [itex]\aleph_2[/itex]?

[graciousgroove]

I know that we can easily construct a set whose cardinality is strictly greater than that of the set of real numbers by taking P(\Re) where P denotes the power-set operator. But as far as I am aware there aren't really any uses for this class of sets (up to bijection), or any intuitive ways of graphically representing them.

Does anyone know of any sets whose cardinalities are \aleph_2, \aleph_3 etc. that have actually been useful in proofs?

I am somewhat math-literate (undergraduate degree in math) but I would really appreciate simple, easily thought-about examples if any exist.

Thanks

 

[micromass]

This is difficult since we don't really know any useful concrete examples of a set of $\aleph_2$ or even $\aleph_1$.

A lot depends on the continuum hypothesis and its generalized version. The continuum hypothesis says that $2^{\aleph_0} = \aleph_1$. The generalized continuum hypothesis is $2^{\aleph_\alpha}=\aleph_{\alpha + 1}$. Both of these statements are perfectly consistent with set theory, but so are their negations. So if you want, you can take the generalized continuum hypothesis as an axiom without running into problems, you can also take its negation.

So, let's say that the Generalized continuum hypothesis is true. Then $|\mathbb{R}| = \aleph_1$. And $|2^{\mathbb{R}}| = \aleph_2$. Are there any interesting sets of cardinality $\aleph_2$ then? I would say yes. A example is the Stone-Cech compactification of the natural numbers. This is important in general topology and other things like C*-algebras.

What if the generalized continuum hypothesis is false. Then it is perfectly possible that $|\mathbb{R}| = \aleph_2$ and this set is of course very important.

It is also possible that $|\mathbb{R}| = \aleph_3$ and so on (although not all alephs are possible, something like $\aleph_\omega$ is not a valid value for $|\mathbb{R}|$, although $\aleph_{\omega+1}$ is).

 

[graciousgroove]

Thanks for your answer micromass. I was assuming the continuum hypothesis (CH) to be true when I made this post; I am familiar with thinking of |\Re| as equal to \aleph_1.

In the case that the CH is true, the example of the Stone-Cech compactification was the sort of thing I was looking for. I tried reading a bit about this example but it is hard to intuit since I'm not familiar with a lot of the set-theoretic terms used in the definition (such as compactification). I remember a conversation I had recently about a set called the "surreal numbers" which is an extension of the reals that includes infinitesimals. Is there an extension of the reals that is somehow similar in construction to the "surreals" that has the same cardinality as the Stone-Cech compactification (Assuming CH)?

 

[mathman]

The set of all real valued functions of real variables is an example. I am not sure if its cardinality is of much interest.