Virgil
Posts:
8,833
Registered:
1/6/11


Re: What is the intuitive meaning of "nonArchimedean" for a valued field?
Posted:
May 14, 2013 2:48 AM


In article <6bb5684c3fc048a18de5cc51f7478a49@a15g2000pbu.googlegroups.com>, mike3 <mike4ty4@yahoo.com> wrote:
> On May 13, 8:17 pm, Virgil <vir...@ligriv.com> wrote: > > In article > > <bf23b5089d6c459bb7975022f1dd0...@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. 

