FOM: surreals and infinitesimals

Neil Tennant neilt at
Mon May 24 21:42:16 EDT 1999

I would like to thank Professor Pais for his informative posting about how
the surreals can be developed.  Not knowing anything about this area of
mathematics, I would like to pose a simple question for which someone on the
fom list might have a ready answer.
Has any theorist of surreal numbers offered [any subclass of] them as an
explication of the concept of an infinitesimal? For any surreal defined on
omega, its extension to surreals defined on ordinals greater than omega
would seem to produce things infinitesimally close to it. (Here, "not
infinitesimally close" can be understood as "differing by a surreal defined
on just omega".)  Thus one could investigate higher-order infinitesimals
by considering ever-greater ordinals as the domains of the surreals in question.

Another thought:

One could take the surreals corresponding to just the rationals; then take
infinitesimals arbitrarily close to those; and *then* perform cuts on the
extended class, to get continuity at a stroke within the class of reals with

Has anyone explored this idea?

If it's standard fare in one of the better-known references, page numbers would
be most appreciated.

Neil Tennant

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