On Sat, Aug 6, 2011 at 6:40 PM, Gavin Wraith
<gavin@wra1th.plus.com> wrote:
In message <4E3D9B31.2030200@interfree.it">4E3D9B31.2030200@interfree.it> you wrote:
> IIRC (too many years since my exams of advanced calculus :-)
> completeness is non-trivial (i.e. interesting from a mathematical POV)
> only in the context of the theory of fields or normed spaces over a
> field (R or C, specifically), or am I mistaken?
Like so many words that maths borrows, 'complete' is very overloaded.
It is a convenient word because it is a verb and an adjective, and
provides a noun 'completeness'. It is used in a huge number of contexts,
and in association with lots of names of individuals who used the word
to some end judged useful by the mathematical community. Just google
and see: Goedel-completeness, Dedekind-MacNeille-completeness,
Stone-Cech completion, ... . In fact most of these uses of the term
can be subsumed under one or two very abstract usages which generalize
the more particular ones. The 'yoga' of completeness goes informally
something like this:
do
local hyperbanana,sit inside,callifragilistic,somebody,X,Y,X'
-- Fit 1
You have some type of gadget - let us call the gadgets 'hyperbananas'.
One hyperbanana may well 'sit inside' another bigger hyperbanana.
Some hyperbananas may have a useful/weird/desirable property of
being 'callifragilistic'.
-- Fit 2
Somebody proves that every hyperbanana sits inside a callifragilistic
hyperbanana.
-- Fit 3
Somebody proves that for any hyperbanana X there is callifragilistic
hyperbana X' with the property that, whatever the callifragilistic
hyperbana Y, if X sits inside Y then X' also sits inside Y.
Under these circumstances one would say that X' was a callifragilistic
completion of X.
return 'completion'
end
--
Gavin Wraith (gavin@wra1th.plus.com)
Home page: http://www.wra1th.plus.com/