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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?

Mike Battaglia
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  • What precisely is the diagram you are thinking of? How do you handle the situation where there is more than one embedding between two ordered fields? – Zhen Lin Nov 10 '22 at 22:33
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    If one uses all embeddings, then you don't have a commutative system, and so the direct limit doesn't make sense. – Joel David Hamkins Nov 10 '22 at 23:38
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    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? – Alec Rhea Nov 11 '22 at 00:07
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    I was imagining when there are several different embeddings, you would choose between them such that the entire thing forms a direct system (e.g. the embeddings can be composed and so on). The different ways of choosing these embedding correspond to different direct systems. So I guess one question is, do these different choices all lead to different direct limits? Or do they all lead to the same direct limit, and is that limit the surreal numbers? – Mike Battaglia Nov 11 '22 at 01:03
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    I am pretty sure that the answer is No. The 'field' of surreal numbers may be identifiable by what Alex wrote, but the isomorphism is not unique, it is not a universal property in the usual sense. – Martin Brandenburg Nov 11 '22 at 01:31
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    The main question is this: every ordered field embeds into this direct limit. Also, every ordered field embeds into the surreal numbers. Are they the same? – Mike Battaglia Nov 11 '22 at 02:39
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    If not, what is the relationship between them? Do we have two non-isomorphic fields which embed into one another? (Do we even know what the cardinality of the direct limit is?) – Mike Battaglia Nov 11 '22 at 02:40
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    I've edited the question to add all of these clarifications. – Mike Battaglia Nov 11 '22 at 02:40
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    Why is this non-unique direct limit appealing to you? What would make it more appealing than the categorical characterization in my last comment? I ask because it seems like the answer to your question is a pretty trivial ‘yes’ in the same non-unique sense as the answer to my linked question was ‘yes’; if this direct limit is more interesting for some reason, I’d be fine working out the details. – Alec Rhea Nov 11 '22 at 03:11
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    Alec, your answer transcends ZFC and even global choice, because of the use of universes. I find it likely that that it is consistent with ZFC that things go badly awry, such as with definable classes only and global choice failing. The universe might not even be linearly ordered – Joel David Hamkins Nov 11 '22 at 03:55
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    @JoelDavidHamkins Interesting; intuitively, do you think this construction ‘should’ be different than the surreals? – Alec Rhea Nov 11 '22 at 05:49
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    My view is that most nontrivial universality properties of the surreals, with respect to proper-class systems (such as direct limited of class directed systems), will require global choice, and not be provable in ZFC alone. – Joel David Hamkins Nov 11 '22 at 13:24
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    @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. – Joel David Hamkins Nov 11 '22 at 13:33
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    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. – Joel David Hamkins Nov 11 '22 at 13:36
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    @JoelDavidHamkins If you think that ZFC+global choice is the best way to formalize this question, rather than putting things into a Grothendieck universe as I had (perhaps naively) done as a suggestion in my post, then I think that is fine and am still curious what the answer is. – Mike Battaglia Nov 11 '22 at 16:03
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    @JoelDavidHamkins Regarding the question of which directed system on ordered fields I am talking about, the question is really this: suppose S is any arbitrary directed system over the class of ordered fields. Is it necessarily true that we always have the surreal numbers as a direct limit? Clearly, as you have shown, there exists one such directed system for which this is true; is it always true? – Mike Battaglia Nov 11 '22 at 16:05
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    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. – Joel David Hamkins Nov 11 '22 at 16:12
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    @AlecRhea thanks for the link, I am still trying to see how it relates to this question. The point of this question was that direct limits also have universally embedding properties so I was curious how these direct limits relate to the surreal numbers. – Mike Battaglia Nov 11 '22 at 16:23
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    @JoelDavidHamkins OK, that all makes sense. So with orders, we may get a "lumpy surreal numbers" instead, with gaps or extra bits or something like that. – Mike Battaglia Nov 11 '22 at 17:13
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    I asked a question about the ZFC version of the question here: https://mathoverflow.net/q/434405/1946 – Joel David Hamkins Nov 11 '22 at 17:18

1 Answers1

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Here is one way to get a positive answer to the title question.

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. $\Box$

I realize that this answer is not achieving the goal you had wanted, which was an alternative route to universality, since I am using universality to prove that the construction has the property of containing copies of all fields.

Joel David Hamkins
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