On May 13, 7:27 pm, William Elliot <ma...@panix.com> wrote: > On Mon, 13 May 2013, mike3 wrote: > > I'm curious about this. The "Archimedean property" for an _ordered_ > > field F means that given any positive elements a and b in F, with a < > > b, then there exists a natural number n such that na < b. > > That is trival. If a < b, then 1a < b and the Archimedean property > always holds. >
Oops. That should be "na > b". :)
> > Intuitively, this means F has no "infinitely big" or "infinitely small" > > elements. > > We could also say that "given any positive element a, then > > there is a natural number n such that na > 1". If the property fails, > > then F contains "infinitely small" elements. > > What does infinitely large and infinitely small mean? >
Usually, "infinitely big" means "bigger than any finite number". If one thinks of the elements 1, 1+1, 1+1+1, etc. as the finite numbers in the field, then if a > 1, a > 1+1, ..., a is "infinitely big". The inverse of an infinitely big element is infinitely small.
> > Now, there is an analogous property for non-ordered, "valued" fields > > (fields with an "absolute value" function added). The "Archimedean > > property" here means that given any nonzero element a e F, that there > > exists a natural number n such that |na| > 1. But what, intuitively, > > does it mean when this property fails? > > That there's some element a, for which the sequence > (|na|)_n is bounded by 1. > > > In that case, there aren't any elements with "infinitely small but > > non-zero" absolute value since the absolute value functions are usually > > taken as real-valued, and the reals are Archimedean (as an ordered > > field). Instead, what happens for such a real-valued absolute value is > > that the triangle inequality strengthens to |a + b| <= max(|a|, |b|) and > > not just |a + b| <= |a| + | b|. This causes the space to behave really > > weirdly(*). But what is the _intuition_ here, and how does this notion > > relate, if at all, to the first one? Especially considering I see in > > papers like this: > > It seems such a norm would give rise to an ultra metric.