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Completions and the Archimedean property August 28, 2009

Posted by David Pierce in Uncategorized.
5 comments

In an ordered Abelian group, one positive element can be described as infinite with respect to another if the former exceeds every integral multiple of the other. If there are no such elements, the group can be called Archimedean.

Non-Archimedean ordered Abelian groups exist: for example, the group of ordered pairs (x,y) of integers, with the left-lexicographic ordering, so that (x,y)<(a,b) if and only if either x<a, or else x=a and y<b.

The main point of this article is to observe that an ordered
Abelian group is Archimedean if and only if it has a completion.
I do not know of a reference for this result, though I can well imagine that a reference exists.

The observation about completeness arises from considering that the field of real numbers is complete in two ways:

  1. It is complete as an ordered field, because every nonempty
    subset with an upper bound has a least upper bound.

  2. It is complete as a valued field, because every Cauchy
    sequence (with respect to the absolute value function) converges.

In sense (2), the field of real numbers is just one example of a
complete field. The complex numbers compose another such field,
as do the p-adic completions of the field of rational numbers.
Every valued field has a completion.

In sense (1) however, the field of real numbers is unique.
I have encountered at least one mathematician who seemed not to
be aware of this, or to have forgotten it, having apparently
confused completeness of valued fields with completeness of
ordered fields.

Possibly the distinction between ordered fields and valued fields
is like that between induction and completion: a distinction that
may be overlooked in one’s early education and then never
returned to.

At the end of his book Calculus, Michael Spivak constructs
the field of real numbers and proves its uniqueness (up to
isomorphism). It was from Spivak’s book that, as a student, I
first learned of the uniqueness of the real field. Spivak
praises the “one truly first-rate idea” in its construction:
Dedekind’s notion of a cut. Yet Spivak is disparaging of
the “drudgery” of going through the details of the construction.

I revisited the construction recently, in a course on
non-standard analysis at the Nesin Mathematics Village. If the
construction of the real numbers was going to be drudgery, I
wanted to see what more general results could be found in the
process.

The Dedekind cut construction gives a completion to every
ordered set (that is, totally ordered set). Indeed, let A be
such a set, and if x is in A, let (x) be the set of elements of A
that are less than or equal to x. Such sets compose a basis for
a topology on A. A cut of A can be defined as a nonempty closed
set in this topology, except A itself, unless this has a greatest
element. Let c(A) be the set of cuts of A. Then:

  1. The set c(A) is ordered by inclusion
    and is complete with
    respect to this ordering.
  2. The ordered set A embeds in c(A) under the map taking x to (x).

Again, an ordered Abelian group is Archimedean if, for any two
positive elements a and b, some multiple of a exceeds b. In
other words, a and b have a ratio in the sense of
Definition 4 of Book V of Euclid’s Elements. Indeed, the
positive part of an Archimedean ordered Abelian group would seem
to be just the set of magnitudes that have a ratio to some given
magnitude: the set is closed under addition, and under
subtraction of a lesser magnitude from a greater.

For any ordered Abelian group A, Archimedean or not, one can take
the completion of the underlying ordered set, and then extend the
definition of addition to the completion. One way to do this is
to define the sum of non-empty proper closed subsets X and Y of A
as the closure of the set of sums x+y, where x is in X and y is
in Y. Then c(A) becomes an Abelian monoid.

However, if A is a not Archimedean, then A cannot be complete,
since if the set of integral multiples of a positive element a is
bounded above by b, then b-a is also an upper bound. In c(A),
the set of multiples of a does have a supremum, c; but then c+a =
c.

Among Archimedean ordered Abelian groups, the group of integers
is the only discrete example, and this is complete. If A is a
dense Archimedean ordered Abelian group, then c(A) is the same.
If A and B are complete dense ordered Abelian groups, with
positive elements a and b respectively, then there is a unique
isomorphism from A to B taking a to b. The idea is that the
ordered field of rational numbers embeds in A under the map
taking 1 to a; then this map extends uniquely to the completion
of the rational field, which is the real field.

Consequently, for every real number a greater than 1, the
additive group of real numbers is isomorphic to the
multiplicative group of positive real numbers under a map taking
1 to a: this is the map commonly denoted by x |—> ax. One
shows that multiplication distributes over addition on the
positive reals, then on all reals, and so the reals compose a
complete ordered field, which must be unique, because the
underlying group is unique as a dense complete ordered Abelian
group.

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