In <hd-dnfiTht7VboLMnZ2dnUVZ_r6dnZ2d@giganews.com>, on 02/16/2013 at 02:59 PM, fom <fomJUNK@nyms.net> said:
>In the logical construction of the real numbers system >using Dedekind cuts, one must fix a choice as to which >kind of rational cut will be taken as canonical.
It's more convenient to do so, but it's not actually necessary.
>> In general , machines can't decide the equality or inequality of real >> numbers ,or infinite strings in general ,without the 0.(9) = 1 >> equivalence of real numbers.
Programs capable of proving theorems have been available for decades.
>So, in the hierarchy of logical definition, one obtains the real >numbers from Dedekind cuts relative to a logical identity >relation. Then, a definition of least upper bound and greatest >lower bound for that system may be defined. Then, provided that >the nature of relations used in the metrization lemma are >satisfiable, one uses the function constructed in that proof to >put a metric on the system of Dedekind cuts.
It's the other way around; one needs the Reals to define a metric, and once one has the Reals the standard metric is a trivial construction, the details of which don't depend on whether one used Axioms, Cauchy sequences of rational or Dedekind cuts of rationals.
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