FOM: Conway's foundational ideas; pointing fingers

[Stephen G Simpson]

My posting of 20 May 1999 15:07:04

 > Presumably the hereditarily countable surreals form a countably
 > complete real closed ordered field of cofinality omega_1, and one
 > could ask if these properties, perhaps with some other elegant
 > properties, suffice to characterize them up to isomorphism.  Such a
 > characterization might be consistent with and independent of ZFC.

If we assume the continuum hypothesis, then general results of model
theory imply that there exists a countably saturated, real closed,
ordered field whose cardinality is that of the continuum, and this
ordered field is unique up to isomorphism.  Is this ordered field also
isomorphic to the ordered field of hereditarily countable surreal
numbers?

(By ``countably saturated'' I mean ``kappa-saturated'' where kappa =
aleph_1, in the familiar sense of model theory, e.g. the Chang/Keisler
textbook.)

Trying to prove this conjecture, I glanced through ``On Numbers and
Games'' some more, but I didn't find where Conway states or proves a
completeness property of the surreal numbers.  Could someone please
help me with this?

Correction:

 >   rank(x) = (sup{rank(y)| y in x}) + 1

should have been

     rank(x) = sup{rank(y)+1|y in x} .

Walter Felscher 21 May 1999 14:31:02

 > It should be observed that it are three fingers pointing at you
 > when you point your finger at somebody else.

Please don't worry, I can stand the heat.  And I presume that
Professor Conway can also stand the heat, else he would not have
called for a Mathematicians' Liberation Movement.

In my opinion, the most relevant criterion for evaluating FOM postings
is not the quantity of heat, but rather something like the ratio of
light to heat.

heat = emotional pressure

light = scientific enlightenment

-- Steve