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

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Posts: 2,396
Registered: 12/8/04
What is the intuitive meaning of "non-Archimedean" for a valued field?
Posted: May 13, 2013 9:32 PM
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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. 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.

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? 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:

"Our usual notions of space and time are built from axioms of
geometry, which
started with Euclid but eventually were formalized by Hilbert. Of
interest to us is the Archimedean axiom, which states that given a
line segment
of any length and a shorter line segment, successive additions of the
short line
segment along the long line segment will eventually surpass the long
line segment. ..."

which we recognize as the "Archimedean property". In Hilbert's axioms,
this Archimedean property seems more related to _ordered_ fields, yet
the paper goes on about non-Archimedean _valued_ fields, namely the p-
adic numbers. How can one intuitively grasp the geometry of a non-
Archimedean valued field and how does it relate to the notion in
Hilbert's axioms?

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