TITLE: Does this construction yield the surreal numbers?
QUESTION [10 upvotes]: There are two simple constructions for creating arbitrarily large non-Archimedean ordered field extensions of the reals.
First given such a field one may consider rational functions over that field with $f(x)$ declared positive if $f(x)>0$
for all sufficiently large $x$.  Secondly one can Dedekind complete the field by filling in all good/narrow cuts in the
field as detailed in 
this discussion.
I would like to know if iterating these two constructions over the ordinals produces the surreal numbers.
Specifically we define $F_0$ to be the real numbers.  For an ordinal $\lambda$ with predecessor $\lambda-1$, define $F_\lambda$ to be the
ordered field obtained by Dedekind completing the rational functions over $F_{\lambda-1}$. Otherwise if $\lambda$ is a limit ordinal,
define $F_\lambda$ to be the union of all $F_\kappa$ with $\kappa<\lambda$.

REPLY [2 votes]: I've written up an expository paper incorporating ideas from this discussion.
Here is a link:
https://drive.google.com/file/d/0B1G4mOmOYMtha1JLc2tTY1d2SnM/view
I welcome any comments, suggestions, etc.