FOM: model theory and ONAG

[holmes@catseye.idbsu.edu]

Pace remarks of Simpson, Conway is well aware that the surreal numbers
are not a new structure -- in the sense of not being a new field.  On
p. 43, Conway says "as an abstract Field, No is the unique universally
embedding totally ordered Field", which strongly suggests that he
knows the model theory (the previous paragraph describes the proof of
uniqueness by a back-and-forth argument).

However, the surreal numbers have a good deal more structure than
that.  The construction {A | B} of a surreal number from sets A and B
of surreal numbers is not definable from field operations.  As a
mathematical structure, the surreals are richer than the unique
universally embedding totally ordered Field; i.e., they are a new
structure after all.  The immediately following sentence on p. 43 is
"We repeat that No has plenty of additional structure which would not
emerge from this "definition" ".

Simpson has been very firm himself about the importance of the details of the
definitions of mathematical structures (dare we recall the discussion
of Boolean rings?).

On p. 30, one can find a description of the representation of the surreals
used by Gonsher, and a proof that it is a representation of the surreals.

I find the whole tone of Simpson's remarks about this book simply astonishing.
This is a wonderful book, and my interest in the foundations of mathematics
was strongly enhanced by reading it as an undergraduate.

On p. 44, we find (in response to someone else's inaccurate remark)
that No is NOT motivated by applications to nonstandard analysis: "So
we can say in fact the Field No is really irrelevant to nonstandard
analysis".  This is firmed up in the following remark that there is no
natural way to get nonstandard models of the integers out of No (and
nonstandard integers are very important in NSA).

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes