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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.

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Conway, ONAG end of Chapter 4, says "The field No is really irrelevant to nonstandard analysis." – Gerald Edgar Mar 19 '12 at 19:49
@GeraldEdgar, the point is not so much that the field No is irrelevant to Robinson's framework but that the field No may be of limited relevance to analysis; see my answer. – Mikhail Katz May 3 at 9:04
up vote 21 down vote accepted

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

“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.

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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 the hyperreals, but even such a simple function as the sine does not extend to the surreals.

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