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Old 2021-04-29, 15:35   #15
Dr Sardonicus
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Feb 2017

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The Archimedean property is that if x and y are in an ordered field, x > 0 and y > 0, there is a positive integer n such that n*x > y.

Assuming the ordered field is the field of rational numbers with the usual ordering, I note that if x and y are positive rational numbers, there is a positive integer M such that M*x and M*y are both positive integers. Then taking n = M*y + 1, we have

n*(M*x) >= n*1 = M*y + 1 > M*y, so that n*x > y.

If we ignore the ordering, and use instead a "non-Archimedean valuation" (p-adic valuation), anything dependent on ordering (like "upper bound" or "least upper bound," and therefore "Dedekind completeness") goes out the window.

Luckily, "Cauchy completeness" (every Cauchy sequence in the field has a limit in the field) can still be used to embed the p-adic rationals (and their extensions) into fields that are (Cauchy) complete WRT a non-Archimedean valuation.

Last fiddled with by Dr Sardonicus on 2021-04-29 at 15:35 Reason: xifgin posty
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