There is often an important difference to be made between "countably infinite" things and "arbitrarily large finite" things. There are so many examples of this that I need not list them all here.

The thing is, "countable infinity" is a cardinal, whereas "arbitrary largeness" is not. However, whatever it is, it does indeed seem to be a unique "size-description" of some kind, which is the unique correct answer to a host of valid, well-formed set theoretic questions. For example:

  • How many nonzero bits can be in the binary expansion of a 2-adic number? "Countably" many
  • How many nonzero bits can be in the binary expansion of a natural number? "Arbitrarily" many

Literally, this is a situation where we can ask a very well-defined question about "how many" there are of a particular object, and get a unique correct answer, yet that answer is not a cardinal number.

So what is it?

It seems to be a "size-description" of some kind, which is smaller than countable infinity, yet larger than any finite number. Is there a way to formalize such "size-descriptions?"

It is noteworthy that "arbitrarily large yet finite" things seem to have the property that if you do something informally "power-set-ish" to them, you get $\aleph_0$. That is, it is almost as if our informal notion of "arbitrarily large" corresponded to something like $\beth_{-1}$. For example:

  • The set of binary sequences of finite length $N$ has $2^N$ elements.
  • The set of binary sequences of "countable" length has $2^{\aleph_0} = \beth_1$ elements.
  • The set of binary sequences of "arbitrary" length has $\aleph_0$ elements.

Note again the use of the term "arbitrary" as a unique, well-formed size-description that is distinct from any natural number, smaller than countable infinity, and has the property that the set of all binary sequences of "arbitrary" length equals $\aleph_0$.

There are plenty of variations of the above; perhaps the most prototypical is to look at the set of all "countable" subsets of $\Bbb N$ (call it $2^\Bbb N$, representing cardinal exponentiation), vs "arbitrarily large finite" subsets of $\Bbb N$ (call it $2^\omega$, representing ordinal exponentiation). Subsets in $2^\Bbb N$ can be "countably large," whereas subsets in $2^\omega$ can be "arbitrarily large" (yet finite); both are valid answers to the question of "how large can they be," but only the first is a cardinal.

It also seems the same principle ought to hold for other cardinals, not just $\aleph_0$ and not even requiring they be well-ordered. For instance, we can likewise look at the set of subsets of $\beth_{\omega}$ of cardinality strictly less than $\beth_{\omega}$. These subsets can be "arbitrarily large yet less than $\beth_{\omega}$", which is a size-description that is not a cardinal.

My question:

Does there exist some system of formally representing the "size-descriptions" presented above? Not all of them are cardinals, however, they do seem to be set-isomorphisms of some kind. Has this been formalized?

  • $\begingroup$ Why not consider $\kappa$ as a label for "arbitrary large yet less than $\kappa$"? Then the cardinality $\kappa$ is represented by $\kappa^+$. $\endgroup$ – nombre Apr 8 '19 at 18:14

Cardinality already does the job, we just have to be careful how we use it.

Generally, when we have a set $A$ of cardinals (and the same idea works for ordinals as well), there are two ways to measure how big $A$'s elements can be:

  • $\sup(A)$, the least upper bound of the elements of $A$.

  • $\min(Card_{>A})$, which is suggestive notation for "the least cardinal $>$ every element of $A$."

In general these are not the same: $\min(Card_{>A})$ is never an element of $A$, while $\sup(A)$ sometimes is and sometimes isn't.

This captures the distinction between "arbitrarily large finite" and "countably infinite." To use your examples, we have:

  • The set $A$ of cardinalities of sets of natural numbers ("finite or countably infinite").

  • The set $B$ of cardinalities of finite sets of natural numbers ("arbitrarily large finite").

Then $\sup(A)=\sup(B)$ but $\min(Card_{>A})=\aleph_1>\aleph_0=\min(Card_{>B})$. The point is:

For talking about how big the elements of a set can be, we get more information by looking at the smallest "too-big" thing than at the upper bound of the sizes of the elements.

Specifically, note that the former determines the latter (if $\min(Card_{>A})$ is a limit cardinal, then $\min(Card_{>A})=\sup(A)$, and if $\min(Card_{>A})=\kappa^+$ then $\sup(A)=\kappa$) but not conversely (if $\lambda$ is a limit cardinal then $\sup(A)=\lambda$ doesn't tell us whether $\min(Card_{>A})$ is $\lambda$ or $\lambda^+$).

Indeed, this is an example of the more general phenomenon that "$<$ is better than $\le$." E.g. in forcing we say $\mathbb{P}$ has the $\kappa$-chain condition ($\kappa$-c.c.) iff every (strong) antichain in $\mathbb{P}$ has size $<\kappa$; think about the difference, using this definition, between (say) the $\aleph_0$ and the $\aleph_1$ chain conditions.

  • Incidentally, the $\aleph_0$-chain condition is silly in the context of forcing. Also, usually we'd write "$\omega_1$-c.c." rather than "$\aleph_1$-c.c." (and etc.). Finally, it's worth noting that "$\omega_1$-c.c." is also called the countable chain condition, and abbreviated "c.c.c."
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  • $\begingroup$ I think that I've hardly ever heard $\omega_1$-c.c. or $\aleph_1$-c.c. being used (at least in contemporary works). But generally speaking, I can't say if I've heard $\omega_\alpha$ or $\aleph_\alpha$ more often (well, usually $\alpha$ is very small, like $2$ or $3$, otherwise we just have $\kappa$ there...). $\endgroup$ – Asaf Karagila May 22 '19 at 7:39
  • $\begingroup$ @AsafKaragila Oh sure, but I wanted to make the point that the language makes sense and supports the "$<\kappa$" flavor. And personally I have seen $\omega$ more than $\aleph$ in this context, although that might just reflect my limited experience. $\endgroup$ – Noah Schweber May 22 '19 at 7:43
  • $\begingroup$ Yes, of course. I'm not disagreeing on that. Just saying what I've heard from my colleagues over the years. And I'm not talking only about the Israeli ones who would have a natural inclination to use $\aleph$... :) $\endgroup$ – Asaf Karagila May 22 '19 at 7:44

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