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?