# Can you build the surreal numbers as a simple direct limit of ordered fields?

The surreal numbers are sometimes called the "universally embedding" ordered field, in that every ordered field embeds into them. What "universally embedding" means seems to be somewhat complicated, so I am curious if a simpler method of building them will do.

The idea is, suppose we take "all of" the ordered fields, with homomorphisms as embeddings between them. We can then build the direct limit, which "all of" the ordered fields embed into. Is the result equal to the surreal numbers?

I put "all of" in quotes because there are clearly a few quirks that are needed to avoid set-theoretic paradoxes. So for instance, instead of looking at "all of" the ordered fields, we can look at only those of cardinality less than, suppose, some strongly inaccessible cardinal $$\kappa$$. Then we can build the direct limit without any problem. Is the result an "initial subfield" of the surreal numbers with birthday up to that inaccessible cardinal. (Or do we get something strictly larger?)

I am sure there are other ways to deal with the set theory issues posed by this question, so I will kind of leave it open to whatever makes for the best answer (as is pretty common when asking questions about surreal numbers in general).

EDIT: a few clarifications

1. Clearly sometimes the choice of embedding from one field into another is non-unique. To phrase the question differently: suppose we have any arbitrary directed system in which the objects are all of the ordered fields. Is the direct limit of this directed system always isomorphic to the surreal numbers?

2. Whatever the direct limit is, every ordered field embeds into it. Also, every ordered field embeds into the surreal numbers. Does this tell us anything about the relationship between the two fields?

• If one uses all embeddings, then you don't have a commutative system, and so the direct limit doesn't make sense. Commented Nov 10, 2022 at 23:38
• I like this question; the surreals are a weakly terminal cogenerator in the category of ordered fields, and this characterizes them ‘universally’ in the sense that any two weakly terminal ordered fields are isomorphic, so the surreals are ‘the weakly terminal ordered field’. Does this answer satisfy you? Commented Nov 11, 2022 at 0:07
• @MikeBattaglia You refer to "the" direct limit of some system, but you haven't defined any directed system of embeddings, and it is not clear that there even is a definable such directed system of embeddings on the class of fields in ZFC, without using global choice. In ZFC, we aren't generally able to make a proper class of arbitrary choices. The surreals are a definable proper class, and to my way of thinking, all the questions here involve quite subtle interaction of the set/class distinction and definability issues, which are unfortunately brushed aside in this discussion. Commented Nov 11, 2022 at 13:33
• I think a natural question here is: can one prove in ZFC that there is a definable directed system of commutative embeddings on the class of all fields whose direct limit is definably isomorphic to the surreal field? I would find this question interesting also just with respect to the order structure. With the global choice, the answers are yes. Without it, I am inclined to expect a negative answer. Commented Nov 11, 2022 at 13:36
• In ZFC alone it is not clear to me that there is a directed commutative system of embeddings on the class of all fields. (My answer doesn't use all fields, but a class of fields containing copies of any given field.) With global choice, this seems fine and one can realize the direct limit. Meanwhile, the uniqueness question is interesting. If we consider just the surreal order (not field), then I believe the answer is negative. One can design a directed system so as to preserve a certain interval as empty. In effect, build the surreals, but only fill gaps outside that interval. Commented Nov 11, 2022 at 16:12

Theorem. There is a definable class $$\mathcal{F}$$ of ordered fields, containing isomorphic copies of any given field, and a directed order $$\unlhd$$ on them, with a definable commutative system of embeddings between them $$\pi_{F,K}:F\to K$$ for $$F\unlhd K$$, such that the direct limit of the system is the surreal field $$\newcommand\No{\text{No}}\No$$.
Proof. The surreal field $$\No$$ itself is definable. Let $$\mathcal{F}$$ be the class of set-sized subfields $$F\subseteq\No$$. Define $$F\unlhd K$$ if and only if $$F\subseteq K$$, and let $$\pi_{F,K}:F\to K$$ be the inclusion map. This is a definable, directed, commutative system of embeddings, and the direct limit is clearly the surreal field $$\No$$ itself, since every surreal number is an element of some set-sized subfield. Every ordered field is isomorphic to a subfield of $$\No$$ by the universality property, and so $$\mathcal{F}$$ contains copies of any given ordered field. $$\Box$$