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.

 

[blue_raver22]

I am not an expert in set theory. However, I can provide some information that may be helpful in understanding sets with cardinalities of \aleph_2 and beyond.

Firstly, the concept of cardinality is a measure of the size of a set, and it is defined as the number of elements in the set. In the context of set theory, the cardinality of a set is related to the concept of the power set, which is the set of all subsets of a given set. The cardinality of the power set of a set with cardinality \aleph_0 (the cardinality of the set of natural numbers) is \aleph_1, and in general, the cardinality of the power set of a set with cardinality \aleph_n is \aleph_{n+1}.

To answer your question, there are indeed sets with cardinalities of \aleph_2, \aleph_3, and beyond that have been useful in mathematical proofs. For example, the Continuum Hypothesis, which states that there is no set whose cardinality is strictly between \aleph_0 and \aleph_1, has been extensively studied and has important implications in set theory and other areas of mathematics.

Another example is the set of all countable ordinal numbers, which has cardinality \aleph_1 and has been used in proving theorems related to the structure of the real numbers.

Additionally, there are many other sets with cardinalities of \aleph_2 and beyond that have been studied in different branches of mathematics, such as topology and functional analysis.

In terms of intuitive ways of representing these sets, it may be challenging as these sets are often abstract and not easily visualized. However, there are some visual representations that can be used, such as the Cantor set, which has cardinality \aleph_1 and can be constructed using simple geometric shapes.

In summary, sets with cardinalities of \aleph_2 and beyond have been extensively studied and have important applications in mathematical proofs and structures. While they may not be easily visualized, they have significant implications in various areas of mathematics.