[HM] Complete ordered fields

Roger Cooke (cooke@emba.uvm.edu)
Sat, 14 Nov 1998 20:26:00 -0000

Many thanks to those who responded to my question about complete ordered
fields. The background of the question was the following. Back in 1962
I took a course out of Hille's <i>Complex Analysis</i> in which the real
numbers were characterized as the unique (up to isomorphism) ordered
field satisfying the Dedekind postulate. The following semester I gave
that definition to Ralph Boas, who was teaching real analysis. He
replied that the Dedekind postulate merely gave completeness, so that
the reals were just a complete ordered field. That set me thinking, and
I later realized that the Dedekind postulate (or, equivalently, the
least upper bound axiom) gives more than just completeness; it also
gives the Archimedean property, as several people here have pointed out.
Some time later I arrived at a counterexample to show that a complete
ordered field need not be Archimedean.

I have posted the counterexample at the following address:

http://www.emba.uvm.edu/~cooke/field.pdf

(You can just click there if you are reading this with your browser.)
I think the example deserves to be better known, as the word "complete"
can be misleading.

Roger Cooke