Are the hyperreals emerging at some stage of the surreal construction?

Question

For me What's the difference between hyperreal and surreal numbers? has a not very satifying answer. I always pictured the hyperreals as some subfield of the surreals naturally emerging when you stop the process of surreal building at some specific ordinal.

Is this a valid thought? If we build the surreals and stop at (for example) $\omega_1$, could it be that these are exactly the hyperreal numbers?

---

Extra:

If we stop building surreals at some limit ordinal $\alpha$, do we always get a field (I do not really think so)? If not for all, are there some ordinals where we get a field? Does this give well-known fields for some interesting ordinals?

Answer 1

$\DeclareMathOperator{\Noo}{No}$

First note that in ZFC, there can be several non-isomorphic fields of hyperreal numbers built as ultrapowers of $\mathbb{R}$ to the power $\mathbb{N}$.

If CH is assumed however, then all those are isomorphic to the field $\Noo(\omega_1)$ of surreal numbers born before $\omega_1$. This is a consequence of the fact that saturated real closed fields of same cardinal are isomorphic, and if $\aleph_1 = 2^{\aleph_0}$, then $|\Noo(\omega_1)| = \aleph_1 = |^*\mathbb{R}|$, where those fields are $\aleph_1$-saturated. I should point out that those isomorphisms are not canonical, which undermines the usefulness of your vision of hyperreal systems as subsurreal systems.

You can easily see that $\lambda$ must be stable under Hessenberg arithmetic (and thus under classical ordinal arithmetic) for $\Noo(\lambda)$ to be a subring of $\Noo$.

If $\Noo(\lambda)$ denotes the subset of $\Noo$ of surreals born (strictly) before $\lambda$, then $\Noo(\lambda)$ is a subfield of $\Noo$ (with nice properties) iff $\lambda$ is a $\varepsilon$-number. This is proven in this paper of Philip Ehrlich and Lou van den Dries.

If the cardinal $\kappa$ satisfies $\forall \lambda < \kappa, 2^{\lambda} \leq \kappa$ (cardinal notations), then $\Noo(\kappa)$ is up to isomorphism the only saturated real closed field of cardinal $\kappa$, because of what I wrote above. What's interesting in realizing it as a surreal subfield is the additionnal structure that you get in a natural way: a simple Hahn series structure, some universal properties, a very nice exponential, formal derivations, all of which aren't obvious consequences of this characterisation. The relation between surreal subfields and fields of functions is still under investigation so to speak, for now there is no description of say $\Noo(\omega_1)$ as a function field but some people are working on it, see for instance here.