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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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Virgil

Posts: 6,983
Registered: 1/6/11
Re: What is the intuitive meaning of "non-Archimedean" for a valued field?
Posted: May 14, 2013 2:48 AM
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In article
<6bb5684c-3fc0-48a1-8de5-cc51f7478a49@a15g2000pbu.googlegroups.com>,
mike3 <mike4ty4@yahoo.com> wrote:

> On May 13, 8:17 pm, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <bf23b508-9d6c-459b-b797-5022f1dd0...@tz3g2000pbb.googlegroups.com>,
> >
> >  mike3 <mike4...@yahoo.com> wrote:

> > > 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.

> >
> > Not quite as stated above.
> >
> > Given 0 < a < b there must be some natural n such that na > b.
> >
> > But if a is negative, one will have na < b for all naturals n.
> >
> > The standard ordered field of reals and all of of its subfields  have
> > the property, but fields with infinitesimal elements do not.
> > --

>
> Correct. I made a mistake/typo: it should be "na > b".


And it should be only for a > 0 and b > 0.
--





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