Can the Surreals be Cauchy complete by allowing for ordinal length sequences? While it’s known that the surreal numbers aren’t complete ($1+1/2+1/4\dots$ approaches a gap instead of $2$), can it have a version of Cauchy completeness by allowing the sequences to be longer as is the case for the hyperreal internal sequences. That is to say, for a given function $s: On \to No$, if $\forall \epsilon \in No^+; \exists N \in On; \forall K>N; |s(K)-L|<\epsilon$, we have a Cauchy sequence which converges? The example give above would then converge to $2$ as expected (or at least I think it does). Is there a problem/improvement to this method and is this the mod natural way to “complete” the surreals or is there no way to do so?
After a comment by @Dave L. Renfro, the question would then be “Do the surreals form a radial space?”
 A: For set-sized ordered fields $(F,+,\times,<)$, the following are equivalent:

*

*$F$ has no proper dense ordered field extension

*every Cauchy sequence $\operatorname{cof}(F) \rightarrow F$ converges, where $\operatorname{cof}(F)$ is the cofinality of $(F,<)$.

*$F$ is complete as uniform space where the uniform structure is derived from the ordering.


We say that $F$ is Cauchy-complete if these conditions hold. There is a dense ordered field extension $\widetilde{F} \supseteq F$ which is Cauchy-complete, and this characterizes $\widetilde{F} / F$ up to unique isomorphism, because it is a final object of the category of dense extensions of $F$ (with commutative triangles as morphisms). It is also initial in the category of cofinal and Cauchy-complete extensions of $F$.
This extension is called the Cauchy-completion of $F$.

The equivalence above works for class-sized fields in NBG set theory, but there need not be a Cauchy-completion for all such fields. Indeed, for surreal numbers, the completion would be "too large to be a class". For instance, one can show in certain conservative extensions of NBG in which $\widetilde
{\mathbf{No}}$ exists that $2^{\mathbf{On}}$ injects in it. This is because every strictly increasing and cofinal sequence $u:\mathbf{On} \longrightarrow \mathbf{On}$ induces a Cauchy sequence $C(u)=(\sum \limits_{\gamma<\alpha}\omega^{-u(\gamma)})_{\alpha \in \mathbf{On}}$ in $\mathbf{No}$, in such a way that no two distinct such sequences may have the same limit in an extension.
One can inject $2^{\mathbf{On}}$ into the class of such sequences $\mathbf{On} \longrightarrow \mathbf{On}$ by sending $v:\mathbf{On} \longrightarrow \{0,1\}$ to $u(v):\alpha \mapsto \alpha+v(\alpha)$ if $\alpha$ is a limit, and $\alpha+1 \mapsto u(v)(\alpha)+ 2^{v(\alpha)}$ for all $\alpha$.
If $\widetilde
{\mathbf{No}}$ were a class in NBG, then it would inject in $\mathbf{On}$ (by the axiom of limitation of size): hence we would have an injection $2^{\mathbf{On}}\rightarrow\mathbf{On}$ as per the conservative extension of NBG, which cannot be.

In particular the surreal numbers themselves are not Cauchy-complete.
