FOM: Conway's surreal numbers

[JSHIPMAN@bloomberg.net]

I shouldn't have said that Conway's construction was of the
"hyperreals", a word which should be reserved for elementarily
equivalent extensions.  Actually, his "surreal numbers" form a
real-closed Field (or a real-closed field if you stop the
construction at the right ordinals).  What I meant was that he
constructed a universe where much analysis could be done via
rigorous use of infinitesimals (in particular results of the
freshman calculus type)--but it is not a conservative extension.
Conway says in his book that "deep logical theorems show that
one could not [get hyperreals] in such a simple way".  It's
easy to see this--Conway's construction doesn't need AC. -- JS