FOM: Conway's foundational ideas; surreal numbers

[Stephen G Simpson]

In my posting of 21 May 1999 14:41:42 I said

 > I conjecture that the hereditarily countable surreals are ...
 > countably saturated as an ordered field.  ... And many ...
 > interesting consequences would ... follow ...

I now think I can prove this.  More generally, let kappa be any
infinite cardinal number.

Theorem.  Let F be a real closed ordered field and suppose that the
ordering of F is kappa-dense, i.e., for every pair of sets X and Y in
F with X < Y and card(X union Y) less than or equal to kappa, there
exists z in F such that X < z < Y.  Then F is (kappa^+)-saturated.

[ This result may be well known to those who know it well.  But I
didn't know it before.  Can anyone supply a reference for it? ]

Proof.  By Tarski's result on quantifier elimination for real closed
ordered fields, any subset of F which is definable over F allowing
parameters from F is a finite union of intervals, all of whose
endpoints are in F.  But then Tychonoff's theorem plus kappa-density
of F implies that any family of kappa such sets has the finite
intersection property.  Hence F is (kappa^+)-saturated.

Let No(kappa) be the set of surreal numbers which are hereditarily of
cardinality less than or equal to kappa.

Proposition.  No(kappa) is a real closed ordered field of cardinality
2^kappa and is kappa-dense.

Proof.  This follow from the arguments in Conway's book ``On Numbers
and Games''.

Corollary.  No(kappa) is (kappa^+)-saturated.

Proof.  Immediate from the proposition and the theorem.

Remark.  The existence of a (kappa^+)-saturated real closed ordered
field of cardinality 2^kappa was already known, as a consequence of
the following general result: If A is any infinite structure, then
there exists a (kappa^+)-saturated structure of cardinality 2^kappa
which has the same elementary theory as A.  (Proof: transfinite
inductive construction of length kappa^+.)  This and other similar
results are to be found in model theory textbooks such as Sacks 1972
and Chang/Keisler 1973.

Corollary.  Assume the GCH at kappa, i.e. 2^kappa = kappa^+.  Then
No(kappa) can be characterized up to isomorphism as the unique
kappa-dense real closed ordered field of cardinality 2^kappa.

Corollary.  Let lambda be an inaccessible cardinal, and let
No(<lambda) be set of surreal numbers which are hereditarily of
cardinality less than lambda.  Then No(<lambda) can be characterized
up to isomorphism as the unique (<lambda)-dense real closed ordered
field of cardinality lambda.

Corollary.  In VNBG with global choice, Conway's ordered field No of
surreal numbers can be characterized up to isomorphism as the unique
set-dense real closed ordered field.

Proof.  Tarski's result implies that any two real closed ordered
fields have the same elementary theory.  Hence the above theorem and
proposition reduce the last three corollaries to special cases of the
following general result: If A and B are lambda-saturated structures
of cardinality lambda and have the same elementary theory, then A and
B are isomorphic.  (Proof: transfinite back-and-forth construction of
length lambda.)  This and other similar results are to be found in
model theory textbooks such as Sacks 1972 and Chang/Keisler 1973.

Remark.  By Tarski's result plus general model-theoretic results, a
lambda-saturated real closed ordered field of cardinality lambda has
defined upon it versions of all the standard transcendental functions
on the reals, e.g. sine, cosine, the exponential function, the Gamma
function, etc.  These transcendental functions are unique up to
automorphism.  By the last three corollaries, these results apply to
each of the surreal fields No(kappa) assuming GCH at kappa,
No(<lambda) assuming lambda is inaccessible, and No itself.  This is
perhaps of interest from the viewpoint of Conway's book, which
contains some comments about analysis on the surreal numbers.

-- Steve