The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: What is the intuitive meaning of "non-Archimedean" for a valued field?
Replies: 11   Last Post: May 15, 2013 4:29 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 2,396
Registered: 12/8/04
Re: What is the intuitive meaning of "non-Archimedean" for a valued field?
Posted: May 13, 2013 10:51 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On May 13, 7:27 pm, William Elliot <> 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.

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.