```Date: May 13, 2013 9:32 PM
Author: mike3
Subject: What is the intuitive meaning of "non-Archimedean" for a valued field?

Hi.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 anatural number n such that na > 1". If the property fails, then Fcontains "infinitely small" elements.Now, there is an analogous property for non-ordered, "valued" fields(fields with an "absolute value" function added). The "Archimedeanproperty" here means that given any nonzero element a e F, that thereexists a natural number n such that |na| > 1. But what, intuitively,does it mean when this property fails? In that case, there aren't anyelements with "infinitely small but non-zero" absolute value since theabsolute value functions are usually taken as real-valued, and thereals are Archimedean (as an ordered field). Instead, what happens forsuch a real-valued absolute value is that the triangle inequalitystrengthens 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 thefirst one? Especially considering I see in papers like this:http://www.math.ucla.edu/~virtanen/website_files/thesis.pdf"Our usual notions of space and time are built from axioms ofgeometry, whichstarted with Euclid but eventually were formalized by Hilbert. Ofparticularinterest to us is the Archimedean axiom, which states that given aline segmentof any length and a shorter line segment, successive additions of theshort linesegment along the long line segment will eventually surpass the longline segment. ..."which we recognize as the "Archimedean property". In Hilbert's axioms,this Archimedean property seems more related to _ordered_ fields, yetthe 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 inHilbert's axioms?
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