# Formally representing “arbitrarily large” as a distinct entity from “countably infinite”

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?

• Why not consider $\kappa$ as a label for "arbitrary large yet less than $\kappa$"? Then the cardinality $\kappa$ is represented by $\kappa^+$. – 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."
• 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...). – Asaf Karagila May 22 '19 at 7:39
• @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. – Noah Schweber May 22 '19 at 7:43
• 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 $\aleph$... :) – Asaf Karagila May 22 '19 at 7:44