Question about the Dedekind completion of a non-archimedean ordered field.
Solution 1:
$\newcommand\Q{\mathbb{Q}}$
It is a great question! In general, the answer is no, for there are fields with elements in the Dedekind closure that do not absorb in this way.
Here is an example. Let $\Q^*$ be a nonstandard version of the rational field, obtained for example as an ultrapower of $\Q$, so that it is an elementary extension of $\Q$. So this is a nonarchimedean ordered field. Furthermore, $\sqrt{2}$ does not exist in $\Q^*$. But $\sqrt{2}$ fills a Dedekind cut $S$ in $\Q^*$ — the cut of elements in $\Q^*$ that are negative or whose square is less than $2$ — and so it is in $D(\Q^*)\setminus \Q^*$. But notice that the cut $S$ has the property that it can be bridged with arbitrarily small elements of $\Q^*$, since this fact is true in $\Q$. In other words, for any $0\lt y$, no matter how small, there is $u<\sqrt{2}$ such that $u+y>\sqrt{2}$. It follows that $\sqrt{2}<\sqrt{2}+y$ for any positive $y$, and so $\sqrt{2}$ is not absorbing.
In any field, one can characterize the absorbing elements exactly by this bridging property. Namely, if $x\in D(F)\setminus F$, then the following are equivalent:
- $x$ is absorbing; that is, $x+y=x$ for some nonzero positive $y\in F$.
- $x-u$ is bounded away from $0$, for $u\lt x$.
- $(x-u)\to 0$ as $u\to x$ from below.
- The cut determined by $x$ is not "bridgeable", that is, it cannot be jumped over with arbitrarily small elements of $F$.
The $\sqrt{2}$ example worked precisely because that cut is bridgeable, and can be jumped over with arbitrarily small elements of $\Q^*$, since the analogue of this is true in the standard model $\Q$. Meanwhile, the cut in your example of the supremum of the finite elements cannot be jumped over even with any finite elements.
Update. Finally, let me say that this bridging idea points out the right way to form the Dedekind completion of a nonarchimedean ordered field: you should let $D(F)$ fill only the bridgeable cuts, rather than all cuts. If you do it this way, you will be able to define the field structure on the completion, and you will thereby produce an ordered field, in which every bridgeable cut is filled, and in which the original field is dense.