Consider the surreal line $\langle\newcommand\No{\text{No}}\No,\leq\rangle$, in its order structure only. This is a proper class linear order, which is universal for all set-sized linear orders, as well as homogeneous and saturated.
In ZFC with the global choice principle, one can prove many further attractive universality features for the surreal line. For example, under global choice the surreal line is universal for all class-sized linear orders. This can be proved by a simple forth argument, just like Cantor's proof about the rational line — by global choice one enumerates the elements of the given linear order in an Ord sequence, and then maps each point to the first-born element in the corresponding cut in the surreals determined by the embedding one has built so far. (It is not clear what happens without global choice — see my question, Is the universality of the surreal number line a weak global choice principle?)
Inspired by an issue I found interesting in the recent question of Mike Battaglia, Can you build the surreal numbers as a simple direct limit of ordered fields?, my question is whether in ZFC we can realize the surreal order as the direct limit of a definable system of embeddings on the class of all linear orders.
That is, define $\mathcal{L}$ as the class of all linear orders. In ZFC can we define a way of fitting these orders together so as to form a coherent directed system of embeddings that realizes the surreal line as the direct limit?
Main Question. In ZFC, is there a definable directed order $\unlhd$ on the class $\mathcal{L}$ of all linear orders and a commutative directed system of embeddings $\pi_{\ell_1,\ell_2}:\ell_1\to \ell_2$ whenever $\ell_1\unlhd\ell_2$, such that the direct limit of this system is definably isomorphic to the surreal line $\langle\No,\leq\rangle$?
By considering how the linear orders embed into the surreal line $\No$, the question can be equivalently asked like this:
Main Question (variation). In ZFC, is there a uniform definable manner of embedding every linear order $\ell$ into the surreal line, in such a way that the images collectively cover the surreal line and any two images are covered by a third?
Such a system of embeddings would induce a directed system on $\mathcal{L}$.
If the global choice principle holds, then the answers are affirmative, since using the global well order of the universe we can definably embed each linear order into the surreals in such a way that they cover the surreals and form a directed coherent directed system.
My question, however, is whether this is possible in ZFC alone, without global choice. I suspect a negative answer. One might keep in mind that it is relatively consistent with ZFC (including the axiom of choice) that there is no definable linear ordering of the universe.
Perhaps one way to prove the negative result will be to prove that in ZFC there is no uniform method of defining embeddings of any given linear order in the surreals. That is, even without the covering and directedness requirements of the revised question, it may be impossible.
Question 2. In ZFC, is there a uniform definable manner of embedding every linear order into the surreals?
I suspect not, since the only way I can see how to do this uses global choice. A negative answer to question 2 would imply a negative answer to the main question (and the variation) as well.
