# Completion of surreal numbers

Surreal number field $$\mathbf{No}$$ is not complete, there are "gaps". Does there exists a completion of it?

I know this question depends on axioms of set theory and more, feel free to assume whatever (consistent) axiom system you want.

My motivation comes from this this question. In short, infinite sums are not possible in $$\mathbf{No}$$ (to my great suprise), but to do measure theory with surreal numbers they should be. Sadly, the formal derivation

$$C = \sum_{n=1}^\infty x = x+\sum_{n=2}^\infty x = x+\sum_{n=1}^\infty x \Rightarrow C = x+C \Rightarrow C=0 \vee x = 0$$

is a big obstacle. But it is crazy that the above infinite sum would be divergent when $$x$$ is sufficiently small, like $$x=\frac{1}{\omega_1}$$. So that gives the question - can completion of $$\mathbf{No}$$ even exist?

• The Cauchy criterion on a sequence $(a_n)_{n \in \mathbb N}$ is equivalent to the sequence being eventually constant - take a surreal $\epsilon > 0$ to be smaller than every nonzero $|a_i - a_j|$. I imagine the same should be true if you were to take generalizations of sequences such as nets. Note that a series is equivalent to its sequence of truncated sums. – Dustan Levenstein Jun 27 at 23:11
• Thanks for this comment. The order completion of No certainly gives new elements, so Cauchy sequences are just not sufficient. Maybe one can show that order completion of No doesn't admit any field structure, but I'm not sure. – mz71 Jun 27 at 23:16
• The least upper bound property can't hold either; if $x$ is an upper bound for $\mathbb N$, so is $x-1$. This is roughly equivalent to the formal derivation you've given in your question. – Dustan Levenstein Jun 27 at 23:17
• You need to define what sort of completion you are asking about. There is the theorem that the reals are the only complete ordered field. The surreals have higher cardinality, so any completion cannot be an ordered field. It seems you should be able to do a topological completion by adding points that are limits of any convergent sequence, but I am way out of my depth here. – Ross Millikan Jun 28 at 1:46
• As a general rule of thumb, the surreal numbers are not good for analysis. That's not what they're built for, and I do not know of even a single instance in which their infinitesimals shed any useful light on analysis. – Eric Wofsey Jun 28 at 2:26