Dedekind Cuts, Fundamental Sequences: why?

What is the step of logic which leads one to seek an extention of the
rational numbers to the real numbers?

I understand the arguments given by Prickert and Görke for all of the
previous extentions. But when they go from the rationals to the reals,
they don't really present a formal equation in need of a solution as they
had for all of the prior extentions. It leave me to wonder what, exactly,
the criteria for success is.
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Relevant Pages

  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... quasi wrote: ... had for all of the prior extentions. ... The rationals don't achieve this -- there are holes. ... of the reals is designed to patch all of the holes. ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... had for all of the prior extentions. ... The rationals don't achieve this -- there are holes. ... of the reals is designed to patch all of the holes. ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... they had for all of the prior extentions. ... The rationals don't achieve this -- there are holes. ... of the reals is designed to patch all of the holes. ...
    (sci.math)
  • Re: Computable functions/reasls: followup.
    ... The computable-function definition above still applies, ... Russian-style constructivism is BISH + MP, ... which were reals that you couldn't tell whether or not were rational. ... special about the rationals; they could be replaced by the integers, ...
    (sci.logic)
  • Re: Cantor Confusion
    ... The "number" pi is definitely a merely fictitious element of continuum. ... naturals, integers, rationals, irrationals, or reals. ... intergers and naturals are genuine. ... genuine numbers to the reals is tempting but not justified. ...
    (sci.math)