Starting from the cardinal $|\Bbb N| = \aleph_0 = \beth_0$, we can generate a larger cardinal in two ways:
- Take the set of all subsets, generating the cardinal $\beth_1$
- Take the set of all well-order-types (up to isomorphism), generating the cardinal $\aleph_1$
I am wondering if, rather than well-order-types, we can take other order-types (up to isomorphism) to generate different cardinals. Are any of these known?
- The set of all total order-types of $\aleph_0$
The set of all partial order-types of $\aleph_0$
The set of all preorder-types of $\aleph_0$
where these are all taken up to isomorphism.
Are the cardinalities of any of these known? Do any lead to another type of cardinal similar to the $\aleph$ or $\beth$ cardinals? How much of AC is required to talk about these?
There's an easy construction which lets us show that there are continuum-many countable ordertypes up to isomorphism: given any infinite sequence of natural numbers $f$, consider the order $$A_f=\mathbb{Z}+f(0)+\mathbb{Z}+f(1)+...=\sum_{i\in\mathbb{N}}[\mathbb{Z}+f(i)].$$ It's easy to check that $A_f\cong A_g\iff f=g$. This immediately implies that all the other numbers are also continuum. Note that we can do this for arbitrary infinite cardinalities: given $f:\kappa\rightarrow\{0,1\}$, let $$A^\kappa_f=\sum_{\eta<\kappa}[\mathbb{Z}+f(\eta)].$$ Again we have $A_f^\kappa\cong A_g^\kappa$ iff $f=g$.
So for every infinite cardinal $\kappa$, there are $2^\kappa$-many isomorphism types of linear orders of size $\kappa$. (And AC was never used at any point in the above.)
Belatedly upgrading my comments below into a new section of this answer, there's a reason why we're not seeing anything other than (at-most-)countable, $\aleph_1$, and continuum when we count isomorphism types of countable linear orders of various natural kinds.
Burgess' Theorem says that every analytic equivalence relation on Cantor space has either (at-most-)countably-many classes, $\aleph_1$-many classes, or continuum-many classes. Now we can in a natural way identify points in Cantor space with linear orders on $\omega$; for any $\Pi^1_1$-in-this-sense set of linear orders $\mathfrak{C}$, the equivalence relation $$A\equiv_\mathfrak{C} B\iff (A\cong B\vee A,B\in 2^\omega\setminus\mathfrak{C})$$ is analytic and essentially counts the number of countable linear orders in $\mathfrak{C}$. Consequently Burgess tells us that every $\Pi^1_1$-definable (in the appropriate sense) class of countable linear orders has either $\le\aleph_1$-many isomorphism classes or continuum-many isomorphism classes. And both "all linear orders on $\mathbb{N}$" and "all well-orders on $\mathbb{N}$" are $\Pi^1_1$ (the latter sharply, the former bluntly).
Incidentally, things get weirder at higher cardinals, but that's a separate issue.