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Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational numbers and real numbers; and the size of information needed to fully describe each rational number and needed to describe each real number.
Almost straightforward variation of this approach uses finite and infinite binary sequences.

Recently I have learned of an even more enlightening picture by a consequence of Baire Category theorem: a fact that every nonempty complete metric space without isolated points is at least of cardinality $\aleph_1$.

Questions:

1) Are there any natural examples that provide an intuitive picture of the size of $\aleph_k$ for $k\in \mathbb{N} : k>1$ ?
2) Are there any theorems about metric spaces (like the one I listed) that would let one visualize the spaces of higher cardinalities?

Asaf Karagila
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1 Answers1

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Cardinality, unlike groups, graphs, trees, or vector spaces, have no structure.

It's just a big bag with the label "This bag contains $x$ many elements." it means that one does not have a standard way visualizing it. Furthermore, there are structures (fields, vector spaces, ordered sets, etc) of every given cardinality. So the bag allows you to dress its content with any given structure.

When approaching to visualize infinite objects, if there is no perspective size it is nearly impossible to see it in a way which can tell you much. If you put $\aleph_1$ many points on a line, you will not see them differently than $\aleph_0$ many points, unless you put an $\aleph_0$ line by the $\aleph_1$ one.

This perspective is even more important once you realize that cardinalities may change via inner models or extension of models, which is very similar to how a pebble and a boulder are nearly invisible from outer space. Using forcing, for example, we can ensure that sets of size $\aleph_1$ in the original model have the same size as sets of size $\aleph_0$ in the new model. This means that we "went further back" and now things of size $\aleph_1$ look so small they are about the same size as $\aleph_0$.

However the above notion is important if you want to consider "absoluteness" of visualization, much like you would like to think that the integers and the natural numbers are somewhat absolute when you imagine them. Once you decide to live in a certain model of set theory, you are saying "I am not moving myself, so the perspective will remain the same".

Personally, I have a way to visualize it. I see large sets, with or without the structures. I find it nearly impossible to describe though, and it is some internal idea which allows me to figure things out (it really does help me more often than you'd think, e.g. to understand exactly what Jensen's Covering Lemma means). Since I work a lot with cardinals which are not $\aleph$-numbers, I usually approach to $\aleph$'s as if they were ordinals.

So I would take the scale that I have chosen ($\aleph_0$, some other countable ordinal, or even much much larger cardinals - perhaps inaccessible or so) and try to stretch lines of the size I want to imagine. If you think about them as stretching into more than one dimension then you can explain how the smaller/larger lines you've taken as a scale won't shrink into nothing-ness.

However the $\aleph$ numbers do have some structure. $\aleph_1$ is exactly the number of countable ordinals. In fact the definition of $\aleph^+_\alpha$ is exactly the cardinality of ways to well order a set of size $\aleph_\alpha$ (up to isomorphism, of course).

Of course this does not apply to limit cardinals, for example $\aleph_\omega$ which is the least $\aleph$ number larger than any $\aleph_n$ for finite $n$. At limit points we simply pick the least possible size, if only to allow some sort of continuity.

The above, regardless to being a natural structure on $\aleph$ numbers, is just one example, and you can easily enough return to the mystery bag with the label "Contain $\aleph_n$ many members!".

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Asaf Karagila
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    Does it make any sense? – Asaf Karagila Dec 07 '11 at 20:40
  • Thanks for your answer Asaf; the part after the horizontal line is still somewhat out of my reach, but I will revisit it once. –  Dec 07 '11 at 21:30
  • @Lovre: I tried to edit it a little bit. – Asaf Karagila Dec 07 '11 at 22:06
  • Yeah, I think I understand it now, after looking up limit cardinals and some other things; thanks for the answer again, I'm going to accept it if no better answer appears in the near future. –  Dec 08 '11 at 11:59
  • @Lovre: If I improve it some more, will that count as a better answer too? :-) – Asaf Karagila Dec 08 '11 at 12:01
  • Sorry if I sounded like I required a better answer; I was just leaving it open for anyone wanting to give any different answers, as I have seen examples of too early acceptance of answers here. But if you have any ideas, feel free to improve it. :-) –  Dec 08 '11 at 12:08
  • I added a a point about forcing, I think it may be confusing - but it is essential in my opinion since alephs are very set theoretic in nature, and the fact is that they may change from one model to another. – Asaf Karagila Dec 13 '11 at 12:33
  • ”Furthermore, there are structures (fields, vector spaces, ordered sets, etc) of every given cardinality.” I know that for you guys finiteness is so trivial a case it hardly deserves mentioning, but a student telling me there are fields of every cardinality would be severely chastised. – PseudoNeo Dec 13 '11 at 14:02
  • @PseudoNeo: I'm not sure what you're trying to suggest. If a student would tell me anything I did not teach him myself will make me question his words. I am not the student here, though. – Asaf Karagila Dec 13 '11 at 14:20
  • @PseudoNeo: Also, I believe that there are ordered sets of every cardinality. Even the empty set, if you're willing to stipulate that the empty set is an ordering on itself. Fields, vector spaces, etc. the point was that you can "dress" a set with some structure and visualize it as such. Be it a field, a linearly ordered set, or a pile of sand. However cardinality in its essence is the lack of structure. – Asaf Karagila Dec 13 '11 at 18:55
  • I'm saying that a finite field has $p^n$ elements, so there are no fields of cardinality, say, 6. I was just amused by your answer: I'm sure the result you state about fields is true for any infinite cardinal but somehow the totally irrelevant case of finite sets is the first one to spring in the mind of the laymathematician (at least to mine). It's a bit like hearing about a theorem about Lie groups, finding it wrong on the first example you try, and submit your remark to the speaker just to hear “Oh yes of course, the theorem is for nonabelian Lie groups...” – PseudoNeo Dec 13 '11 at 23:30
  • And once again, I'm crashed under the burden of having to write in a foreign language: I'm strictly incapable of explaining that something makes me laugh, or why. (Of course, it is still possible I misunderstood this particular sentence of yours entirely). – PseudoNeo Dec 13 '11 at 23:33
  • @PseudoNeo: I see. I did not mean, of course, to ignore the finite case; I just think that it is (1) easy to visualize finite sets; and (2) you can impose other sort of structures on the set. I mean, how would you go and imagine the $p$-adic numbers, even though they form a field they are immensely harder to visualize. Luckily to imagine a set of size continuum we can just visualize the real line, or a 3D-space... – Asaf Karagila Dec 13 '11 at 23:33