FOM: surreal numbers

[Stephen G Simpson]

John Pais 24 May 1999 18:50:34

 > Steve, it's a pity that you found it necessary to create that
 > nonexistent quote below. ...

Wait a minute.  Are you saying that your quote ``the enrichment of
mathematics by the inclusion of a new structure with interesting
properties'' did *not* come from Harry Gonshor's book ``Introduction
to the Theory of Surreal Numbers'', CUP, 1986 ???

In the meantime I noticed that the characterization of the surreals in
terms of transfinite sequences of +'s and -'s also appears in Conway's
1976 book ``On Numbers and Games''.  This is Conway's notion of ``sign
expansion'', page 30.  From your 24 May 1999 18:50:34 posting, it
appears that Gonshor 1986 reworked the theory using this as the
*definition* of the surreals.  Is that correct?

I haven't looked at Gonshor's book.  Does Gonshor offer any
amplification or discussion of Conway's *foundational* views (the
stuff on pages 64-67 of ``On Numbers and Games'')?

A *mathematical* question that might be interesting to discuss here on
FOM is: What additional information is given by the Conway 1976 or the
Conway/Gonshor 1986 construction of saturated real closed fields,
above and beyond the more general model-theoretic construction (dating
from the late 1950's I think)?

It seems to me that the Conway and Conway/Gonshor constructions do
give additional insight, because they are more concrete and explicit.
In particular, the general existence and uniqueness theorems for
saturated models use the axiom of choice, but the Conway and
Conway/Gonshor constructions do not.

Does anyone want to discuss this?  Pais?  Shipman?

John Pais 24 May 1999 18:50:34 quotes from Gonshor [ ??? ]

 > Finally, as a culmination of the results of this chapter we have
 > shown that the subset of surreal numbers a such that |len(a)| <= d
 > for any fixed infinite cardinal d is a real closed field. ... These
 > are all "honest" fields since their carriers are *sets*.

In the notation of my posting of 21 May 1999 19:59:44, this is just
No(d), i.e. the field of Conway 1976 surreals which are hereditarily
of cardinality less than or equal to d.  I characterized it up to
isomorphism as the unique (d^+)-saturated real closed field of
cardinality 2^d, assuming GCH at d.

 > The field of all surreals of countable length should be a
 > worthwhile object for further study."

This is just No(aleph_0).  In 17 May 1999 21:15:38 I referred to this
as the hereditarily countable surreals.  Shipman didn't like it much,
but I still like it.  I still don't see anything in Conway's book that
wouldn't make just as much sense when restricted to No(aleph_0).

-- Steve