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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?

mz71
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    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 '20 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 '20 at 23:16
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    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 '20 at 23:17
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    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 '20 at 01:46
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    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 '20 at 02:26
  • @DustanLevenstein: What about the Cauchy criterion on an “class length sequence” $(a_n)_{n\in\mathrm{On}}$? – celtschk Jun 28 '20 at 11:01
  • You just showed that there is no coherent notion of sum which 1) satisfies some compatibility with finite sums 2) makes one non-zero constant sequence summable, with non-zero sum. Your argument is valid in any group. How would the surreal numbers help? – nombre Jun 29 '20 at 10:43
  • @nombre If we could weaken the axioms, maybe it would help. The ordinal number arithmetic is really weak algebraically, but powerfull. Things like ω-sum of 1/ω = ω*1/ω = 1. We could maybe define infinite sums as generalized product like the above. – mz71 Jun 29 '20 at 10:47
  • @RossMillikan It may not be a field, mind you. Ordinal numbers are not a ring, so the completion of surreal numbers could not be a field. – mz71 Jun 29 '20 at 10:49
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    But this is just multiplication. What axioms would you choose? It makes more sense to ask a question such as "is there a summation operator such that this and that property holds" than to ask whether mixing keywords would solve a problem that is not stated. – nombre Jun 29 '20 at 10:54

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Analysis on Surreal Numbers mentions the Dedekind completion $^$, basically identifying the gaps as new elements in this collection. Then we can define things like $∞=\{ℝ|\text{all positive infinite surreals}\}$, which is not a surreal number since the right side is a proper class, and, conversely $\frac{1}{∞}=\{0|\text{all positive surreals}\}$. While before the surreals could be defined as ordinal-sized sequences of $+$ and $-$, the gaps are sequences of length $$.

This construction brings back some useful analytic properties, but it is not a field anymore.

IS4
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