FOM: surreals and infinitesimals

Stephen G Simpson simpson at
Wed May 26 13:16:29 EDT 1999

Neil Tennant 24 May 1999 21:42:16

 > Has any theorist of surreal numbers offered [any subclass of] them
 > as an explication of the concept of an infinitesimal? 

Conway touches on infinitesimal surreal numbers on page 12 of ``On
Numbers and Games''.  Beyond this, I don't know what more Conway and
his followers have done with infinitesimal surreal numbers, because I
haven't read their writings.

However, other people have certainly done a lot with infinitesimal
surreal numbers, in another guise.  There is a vast theory of
nonstandard analysis, starting with Abraham Robinson's rigorous
version of Euler's infinitesimal calculus, and further developed by
H. J. Keisler and many others.  Nonstandard analysis has gotten a lot
of play and has many interesting applications to analysis, etc.  

The techniques of nonstandard analysis are based on model-theoretic
constructions, specifically saturated real closed fields with
additional structure.  (The term ``hyperreals'' is used.)  By my FOM
posting of 21 May 1999 19:59:44, saturated real closed fields are the
same thing as surreal numbers.

Does anybody know whether either of these two groups (Conway et al,
Keisler et al) has ever acknowledged the existence of the other?

-- Steve

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