Der Zahlbegriff, Math Reviews 84f:04001; translated as
The concept of number, Math Reviews 90c:04001
The first couple of chapters of Artmann's book deal with the real
numbers; later chapters cover some other fields too. Artmann
surveys several different notions of "completeness," and shows which
ones are equivalent and which ones aren't. I don't have a copy
of the book here, but I have notes about some of the results.
The least-upper-bound property is what I usually call
"Dedekind complete". There exists a Dedekind complete ordered
field, and it is unique up to isomorphism; thus we can call
it "the" Dedekind complete ordered field -- i.e. the reals.
Dedekind complete implies Archimedean.
Call a sequence x_n "Cauchy" if, for each positive member r
of the ordered field, there is some M such that m,n>N implies
-r < x_m - x_n < r. Say the ordered field is "Cauchy complete"
if each Cauchy sequence is convergent in the order topology.
Then it can be shown that an ordered field is Dedekind
complete if and only if it is both Cauchy-complete and
Archimedean. There are lots of examples of fields that
are Archimedean but not Cauchy complete (for instance,
the rationals). One of Artmann's exercises gives an example
of an ordered field that is Cauchy-complete but not
Archimedean. If I recall correctly, it is very similar
to Roger Cooke's example, but it is formulated in terms
of some sort of power series.
Artmann says that his treatment is based on
Steiner, Hans-Georg
Aequivalente Fassungen des Vollstaendigkeitsaxioms fuer die
Theorie der reellen Zahlen.
(German) Math.-Phys. Semesterber 13 1966 180--201.
Math Reviews 34 #2461
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Eric Schechter *** http://www.math.vanderbilt.edu/~schectex/
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