Another place where ordinals are found is in simplicial sets, which are Set valued functors from a category Ord of finite ordinals and order preserving functions. Think of real geometric n-simplices on n vertices, like the tetrahedron for the ordinal 3. But once again, in defining Ord, we assumed that we knew what an ordinal was. The only way out of this dilemma seems to be to construct ordinals as we define n-logoses. For an elementary topos, for example, we certainly need the concepts 0 and 1 (empty set and one point set), but the concept of the ordinal 2 could wait until we need it for the triangle of ternary logic. This suggests using this triangle whenever we might be tempted to use the ordinal 2. For example, a simplicial category Ord can't be defined until we have a notion of $\omega$-logos. Sounds like an awful lot of work!
The ancients knew that for a set theoretic cardinal n, there was always a decomposition into prime factors. This is the fundamental theorem of arithmetic, but it does not hold for general fields. Anyway, there's a problem with Euclid's original proof by contradiction: it's not constructive. Just because a statement is false, doesn't mean we can assume that its converse is true! And what is a converse? We've seen that 3-logoses require a notion of ternary complementation. Somewhere in the land of $\omega$-logoses there is an awfully complicated notion of primeness.
If any of this makes sense, it suggests that the infinite branchings of the surreal tree follow the process of logosification. Looking at the positive half only, we see a simple Y tree at the level of 2. At the branches, the number 2 has been defined along with the number 1/2. We conclude that the concept of 2-category must be accompanied by a T-dual concept of 1/2-category!
Maybe the appearance of associahedra as coherence laws for n-categories would be a little less surprising in this setup. That is, an (n+2)-leaved 1-tree polytope appears in the definition of n-category, as n goes to infinity.