TITLE: Surreal numbers vs. non-standard analysis
QUESTION [26 upvotes]: What is the relationship between the surreal numbers and non-standard analysis?
In particular, is there a transfer principle for surreal numbers they way there is for NSA?
A specific situation in which such a transfer principle would be useful arose in the thread Uniformizing the surcomplex unit circle ; can the surjectivity of the map $t \mapsto e^{it}$ from the reals to the complex unit circle be transferred to the surreals?  Presumably, one would need a definition of the map that was in some sense first-order; what sorts of definitions count as first-order?  It is not clear to me how definitions involving the two-sided bracket operation can be fit into a first-order framework.

REPLY [6 votes]: Coming back to the first post. Most of modern mathematics is set-theoretic, that is, it studies sets of different kind, so that reals, real and complex functions, relations on reals, as well as a variety of more complex objects like the Hilbert space - are sets of this or another kind. In that sense, any mathematical definition is a 1-st order one, assuming there is no restriction on using the language of set theory within the common axiomatics.
Regarding the surs. The definition of them yields a certain ordered field, perhaps maximal in some well-defined sense, and nothing more. That surs are so attractive to some kind of mathematically-complying minds is, in my opinion, explainable that this still is a very rare domain where meaningful facts can be explored or observed, rather than proved. On the other hand, the students of surs I believe cannot care less about some transfer and about whether their omni-something does not satisfy some Peano axiom. After all, p-adic numbers do not satisfy Peano axioms either, but who cares. 
Further it happens that the surs are isomorphic (in a class theory) to a certain nonstandard universe, defined by totally different means and towards quite different goals. This allows to enrich the surs by a variety of constructions (like the sine function) beyond their native field structure. In this case, a devoted student of surs might be interested to really figure out in some strict, well defined terms, whether a consistent sine function can be defined on surs by pure sur-means. For instance, consider a version of NBG which proves the existence of surs as a class but is not strong enough to prove the mentioned isomorphism, and prove that such a theory does not imply the existence of a consistent sur-sin. This can be very complex though.