Surreal numbers vs. non-standard analysis 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.
 A: 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.
A: In the final section of my paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small” (The Bulletin of Symbolic Logic 18 (2012), no. 1, pp. 1-45, I not only point out that the real-closed ordered fields underlying the hyperreal number systems (i.e. the nonstandard models of analysis) are isomorphic to initial subfields of the system of surreal numbers, but that the system of surreal numbers itself is isomorphic to the real-closed ordered field underlying what may be naturally regarded as the maximal hyperreal number system in NBG (von-Neumann-Bernays-Gödel set theory with global choice)—i.e., the saturated hyperreal number system of power On, On being the power of a proper class in NBG. It follows immediately from the latter that the ordered field of surreal numbers admits a relational extension to a model of non-standard analysis and, hence, that in such a relational extension the transfer principle does indeed hold.
By the way, by an initial subfield, I mean a subfield that is an initial subtree. Discussions of surreal numbers (including most of the early discussions) that downplay or overlook the marriage between algebra and set theory that is central to the theory overlook many of the most significant features of the theory. In addition to the paper listed above, this marriage of algebra and set theory is discussed in the following papers which are found on my website http://www.ohio.edu/people/ehrlich/
“Number Systems with Simplicity Hierarchies: A Generalization of Conway’s Theory of Surreal Numbers,” The Journal of Symbolic Logic  66 (2001), pp. 1231-1258. Corrigendum, 70 (2005), p. 1022.
“Conway Names, the Simplicity Hierarchy and the Surreal Number Tree”, The Journal of Logic and Analysis 3 (2011) no. 1, pp. 1-26.
“Fields of Surreal Numbers and Exponentiation” (co-authored with Lou van den Dries), Fundamenta Mathematicae 167 (2001), No. 2, pp. 173-188; erratum, ibid. 168, No. 2  (2001), pp. 295-297.
A: The real question as far as "ordinary mathematics" is concerned is whether there is a set-size surreal extension of the reals useful in doing analysis, and that as a very minimum admits a sine function. As far as I know the answer is negative.
Namely, there is no transfer principle in the surreals other than the one transfered from the hyperreals.  Therefore it one wishes to do analysis with anything smaller than the absolutely largest class of numbers, the surreals are not an option.  For example, all real functions extend to hyperreal extensions of the real field, but even such a simple function as the sine does not extend to surreal extensions (without passing via an identification of a maximal class-size surreal field and exploiting an identification of the latter with a class-size hyperreal field and importing a hyperreal transfer principle via the identification).
