> 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.
> 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?
> 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.