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Topic: No John Gabriel, an infinite sum is a limit and 0.333... is indeed
1/3. Plus several other misconceptions Gabriel has

Replies: 2   Last Post: Oct 4, 2017 2:58 PM

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bursejan@gmail.com

Posts: 4,572
Registered: 9/25/16
Re: No John Gabriel, an infinite sum is a limit and 0.333... is
indeed 1/3. Plus several other misconceptions Gabriel has

Posted: Oct 4, 2017 2:58 PM
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Remark: It would surprise me, if there were some
circular constructions of the reals floating around.

Circular is not necessarely bad, for example if you
add some minimality criteria to a montone operator,

you might get unique least fixpoints, etc.. etc..
Here is a paper discussing reals and circularity:

Comparing inductive and circular definitions:
parameters, complexity and games
http://www.illc.uva.nl/Research/Publications/Reports/PP-2004-07.text.pdf

But maybe there is somewhere a more down to earth
paper, that shows a inductive real construction?

Is this possible? Does Cauchy or
Dedekind have an inductive aspect?

Am Mittwoch, 4. Oktober 2017 20:46:01 UTC+2 schrieb burs...@gmail.com:
> Yes, this wouldn't work somehow, it
> would be circular, not foundational:
>
> _
> / \
> real |
> / \_/
> rational
>
> Running into case 1 or 2 of the
> Münchhausen trilemma.
>
> Am Mittwoch, 4. Oktober 2017 20:41:08 UTC+2 schrieb Markus Klyver:

> > And no, we don't define real numbers as limits of Cauchy sequences. That's an other strawman and completely illogical. We can, however, define a real number as an equivalence class of rational Cauchy sequences. An equivalence class is not a limit. An equivalence class is a set of equivalent elements under a certain equivalence relation.




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