Are Surreal Numbers the same as Trans-series? I recently found the paper of Berarducci + Mantova [1, 2] saying that surreal numbers are equivalent to trans-series.  These are very different objects:


*

*trans-series are used in physics to correct, Laplace transforms [3]

*Surreal Numbers, originate in Logic and describe combinatorial game theory, but may be used in Analysis [4].


Has anyone checked this equivalence?  Is it correct? 
 A: It seems there might be some confusion about the term "equivalent".  The two structures in question are "equivalent" in a technical sense, i.e. elementarily equivalent (in the language of ordered differential rings). 
Let $\mathcal{L_d} = \{+,\times,0,1,<,\partial\}$ be the language of ordered differential fields.  The field of Trans-series is naturally an "ordered differential field". A priori, the Surreal numbers are an ordered field and not naturally a differential field. On the other hand, Berarducci and Mantova constructed a formal differential operator over the Surreal numbers (which allows one to view the field as a differential field). 
In the paper The Surreal Numbers as a Universal H-field (as pointed out above by nombre), the authors show that the two structures are elementarily equivalent as ordered differential fields (Actually, they show something much stronger. They show that the field of Trans-series is an elementary substructure of the field of Surreal numbers). In particular, for any sentence $\varphi$ in the language $\mathcal{L_d}$, we have that $\mathbf{No} \models \varphi \iff \mathbb{T} \models \varphi$, i.e. for any sentence in this language, it is true in the (ordered -differential) field of Surreal numbers iff it is true in the (o.d.) field of Trans-series. 
To reiterate: These results only applies to first order sentences in the language of ordered differential rings. This does not apply to second order properties, or non-trivial extensions of these languages.   
