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I'm looking for a concrete example of a complete (in the sense that all Cauchy sequences converge) but non-archimedean ordered field, to see that these two properties are independent (an example of archimedean non-complete ordered field is obviously the rationals).

Thank you in advance.

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    Complete in what sense? Cauchy? Non-standard Cauchy (that is Cauchy sequences which might be longer than $\omega$ but with $\epsilon$ from the field itself, and not real/rational as usual)? Is it complete in the form of order completeness, namely Dedekind complete?2011-01-16
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    I noticed that complete could mean different things and edited. I'm refering to (ordinary) Cauchy completeness.2011-01-16
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    The completion of any non-archimedean ordered field is still non-archimedean.2011-01-16
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    For a general ordered field, I don't think Cauchy sequences give the appropriate notion of completeness; I think you want Cauchy nets or filters.2011-07-24
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    @HarryAltman Why do you say that?2016-05-10
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    I mean, that *is* the notion of completeness used in the theory of uniform spaces (which an ordered field naturally is). Sequences just aren't enough to capture the toplogy/uniform structure in general. If your field is first-countable, then sure, fine, sequences should suffice. But in general you need nets or filters. You *could* consider "sequence-completeness", I suppose, but I don't see why you'd want to.2016-05-10

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