Constructing the surreal numbers as iterated Hahn series
A theorem due to N. Alling (Foundations of Analysis over Surreal Number Fields, §6.55) states that the surreal numbers are isomorphic, as an ordered and valued field, to the field of Hahn series with real coefficients and value group the surreal numbers themselves. There is also a restricted version, which I'll refer to in order to avoid the (IMHO uninteresting) foundational difficulties in dealing with classes: if
Upon reading this, I thought to myself, “well, this is nice, this means the surreal numbers can be given a construction as iterated Hahn series, something along the lines of: start with the reals, take the Hahn series over that, then take the Hahn series over that (as value group), repeat transfinitely, and voilà, surreal numbers”. Unfortunately, it seems I was being a bit naïve there.
Let us define
Thus, my question is: is there some variant of this construction that will succeed in constructing