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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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mike3

Posts: 2,396
Registered: 12/8/04
Re: What is the intuitive meaning of "non-Archimedean" for a valued field?
Posted: May 14, 2013 4:27 AM
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On May 13, 11:48 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> <6bb5684c-3fc0-48a1-8de5-cc51f7478...@a15g2000pbu.googlegroups.com>,
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>  mike3 <mike4...@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.
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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.

>
> > > Not quite as stated above.
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> > > Given 0 < a < b there must be some natural n such that na > b.
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> > > But if a is negative, one will have na < b for all naturals n.
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> > > 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.
> --


I already mentioned that a and b should be "positive elements" (i.e.
greater than 0).



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