Description of largest possible countable set / number

Question

I am looking for an elegant / standard (if any) description of the largest countable set.

A first naive approach would be to construct this set, X, by taking the integers (0 to, but not including, ω_0, the smallest "infinite" number / ordinal - denoted here by the limit 0 -> ω_0 ) and repeatingly applying the function f x = x * (0 -> ω_0) an (0 -> ω_0) amount of times.

But this would then end us up with X = (0 -> ω_0) ^ (0 -> ω_0), which we can make larger by repeatingly applying f x = x ^ (0 -> ω_0) an (0 -> ω_0) amount of times.

So this hints at a recursive description of abstract operators where we can "upgrade" the operator (0 -> ω_0) amount of times. The set X would then be constructed by applying this "last" operator (described by the above limit, i am well aware the exact last operator cannot exist), on the Natural numbers (0 -> ω_0).

As you can see from this post de description easily becomes quite messy. Many introductory resources show this messiness with amusement and leave it at that, whereas the more serious resources are densely packed with many different terminologies while i just want to use a simple description of this X without having to learn all concepts surrounding it. Can you help me?

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P.S. Should i be using number or amount when talking about (0 -> ω_0)? A limit is not a number i would think..

P.P.S. Is there a more standardized way of writing (0 -> ω_0)?

(Edit) P.P.P.S. Maybe i am not so much asking about how to construct the set X, but more-so about the largest possible number constructable through countable means, phi, such that i can describe X as (0 -> phi).

E.G. : take the operator "@" to be the operator obtained by "upgrading" the operator "*" up to (limit of ω_0) times. The largest possible "number" phi would then be the (limit of ω_0) @ (limit of ω_0). phi might be an uncomputable number, which is fine by me. Maybe i can describe phi as (ω_1 - 1) and (limit of ω_0) as (ω_0 - 1).

Does this make any sense? Can you give some pointers on how i can describe my question better?

Answer 1

Your question is unclear for multiple reasons; it's not clear what you mean by "largest," and it's also not clear whether you're talking about sets / cardinals, or ordinals, or numbers; all of these are different. A precise technical definition of "largest" is the following: in any partial order, a largest element or greatest element is an element $x$ such that $y \le x$ for all other elements $y$. You could be talking about one of three (at least) partial orders:

1. Sets ordered by inclusion. In this case "largest countable set" would mean a countable set which contains all other countable sets as subsets. There is no such set; the union of all countable sets is so large it is not even a set.

2. Cardinals ordered by the existence of an injection. In this case every countable cardinal has the same size, so any countable cardinal is the "largest" countable cardinal.

3. Ordinals ordered by initial segment. In this case "largest countable ordinal" would mean a countable ordinal which contains all other countable ordinals as initial segments. There is no such ordinal, because if $\alpha$ is any countable ordinal then the collection of all ordinals $\le \alpha$ is itself a countable ordinal strictly larger than $\alpha$.

4. "Numbers," whatever that means. But then I have no idea what you mean by a "countable number."

You can describe a bunch of countable ordinals by starting with $\omega$ and then, for example, iterating to get $2 \omega, 3 \omega, 4 \omega, \dots$. Doing this countably many times gets you $\omega^2$. Then you can consider iterating $\omega^3, \omega^4, \dots$. Doing this countably many times gets you $\omega^{\omega}$. Then you can consider iterating $\omega^{\omega^{\omega \dots}}$. Doing this countably many times gets you an ordinal called $\epsilon_0$. Then you can keep going from here. Any specific ordinal notation for writing down countable ordinals eventually exhausts itself, because it can always be diagonalized over by considering the supremum over all ordinals expressible via that notation (of which there are countably many!), so at some point you always need new notation to write down even larger countable ordinals.

You may be interested in reading Scott Aaronson's classic Who Can Name The Biggest Number? as well as this related blog post. There are interesting connections between naming big numbers, naming big countable ordinals, and naming fast-growing functions; in some rough sense the three problems are equivalent.