About ordinals and cardinals in structural set theory Among the most important concepts of set theory for mathematical real life applications are ordinal numbers and cardinal numbers. In material set theory, ordinal numbers are defined as transitive sets, well-ordered by the membership relation. It can be shown that every well-ordered set is order isomorphic to a unique ordinal number. In categorical terms, ordinal numbers form a skeleton of the category of well-ordered sets. By the well-ordering theorem, every set admits a well-ordering, whence is isomorphic to the underlying set of an ordinal number. Choosing for each isomorphism class the smallest such ordinal number, we obtain a skeleton of the category of sets given by the cardinal numbers. Note that by a sufficiently strong version of global choice, the existence of skeletons is immediate.
Now the categorical minded reader might feel uncomfortable, since quite a lot "evil" is going on here. First of all, working in structural set theory like SEAR or ETCS, there is no such thing as a membership relation on a set. Secondly, skeletons are not really of use and most often, it is better to just prove theorems for all objects of a given category instead of restricting one's attention to representatives for each isomorphism class. Finally in structural set theory it is not even possible to express equality of abstract sets, hence uniqueness of the ordinal number associated to a well-ordered set.
So the following questions arise: Do we need to know a relatively concrete description of representatives for isomorphism classes in $\mathbf{Set}$ at all? Do we need material properties of those, like being a transitive set in set theory itself or elsewhere? Is it possible to recover the theory of transfinite induction in structural set theory, replacing the partial order of ordinal numbers by the preorder aka thin category of well-ordered sets? Where do the technical details of developing the theory of ordinal numbers in material set theory move? Perhaps most of them disappear or are related to proving that well-ordered sets are "well-preordered" by the initial segment-relation? Are there philosophical issues when interpreting something like $\dim V=\aleph_0$ as "there is a basis of $V$, which is isomorphic to $\mathbb{N}$" rather than "the mathematical object $\dim V$ is equal to the mathematical object $\aleph_0$"? Perhaps the first interpretation is more natural anyway?      
 A: To answer the more "philosophical" part: It is very useful to have a skeleton of $\mathsf{Set}$. First recall that natural numbers were invented to form a skeleton of $\mathsf{FinSet}$. Humans wanted to compare finite sets of objects in real life with each other, and it was useful to have a criterion when two finite sets are (what we now call) equipotent. This remains true for not-so-real-life objects in mathematics, and it remains true for non-finite sets. According to your profile ;-), you are more interested in algebra, so let me mention as an example the Ulm invariants which are cardinal numbers which classify certain torsion groups.
(Since you have asked a lot of questions, it could be a good idea to ask them separately, if you are still interested in them.)
A: In the Homotopy Type Theory Book, a more structural approach to define cardinal and ordinal numbers is presented. See Chapter 10:

  
*
  
*The type of cardinal numbers is the 0-truncation of the type Set of sets:
  $\mathrm {Card} :≡ ||\mathrm{Set}||_0$
  
*An ordinal is a set $A$ with an extensional well-founded relation which is
  transitive, i.e. satisfies $∀(a, b, c : A).(a < b) → (b < c) → (a < c)$.
  

