Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.
|
|
|
|
Re: distinguishability - in context, according to definitions
Posted:
Feb 18, 2013 10:27 PM
|
|
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.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spamtrap@library.lspace.org
|
|
|
|
