Re: Well Ordering the Reals

In article <MPG.1dd56f83edcb67c698a653@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> Jesse F. Hughes said:
> > stevendaryl3016@xxxxxxxxx (Daryl McCullough) writes:
> >
> > > Jesse F. Hughes says...
> > >>
> > >>stevendaryl3016@xxxxxxxxx (Daryl McCullough) writes:
> > >>
> > >>> Tony Orlow says...
> > >>>
> > >>>>How would one prove that it contained all reals?
> > >>>
> > >>Daryl, with all due respect, I found your response odd.
> > >>
> > >>> To prove that a set contains all real numbers, you need to
> > >> ^^ One way to...[1]
> > >
> > > Right. One way.
> > >
> > >>> show that it contains all rational numbers, and then show
> > >>> that your set is complete. To be complete means that for
> > >>> every Cauchy sequence of reals in your set converges to a
> > >> ^^^^^ rationals
> > >
> > > No, to say that the space is complete is to say that
> > > *every* Cauchy sequence converges, not just Cauchy sequences
> > > of rationals. Of course, it amounts to the same thing, because
> > > for every Cauchy sequence of reals there is a Cauchy sequence
> > > of rationals converging to the same thing.
> >
> > Well, with your acceptance regarding "one way" above, I don't have a
> > real argument here. But your requirement is still apparently
> > overstated. My version is better, in my humble opinion.
> >
> > >
> > >>> real in your set.
> > >> ^^^ element (which is necessarily real)
> > >
> > > No, to prove that a set contains all the reals doesn't
> > > require proving that it contains *only* reals. It might
> > > have other objects that are not reals.
> >
> > Right, but Tony has a set. It includes Q. He wants to know that it
> > includes R. It is sufficient to prove that every Cauchy sequence of
> > rationals converges to an element in Tony's happy set.

It also suffices to show that for every non-empty set of TO-numbers
which is bounded above, there is a TO-number which is the least upper
bound of all TO-numbers for that set.
> >
> > One can *then* conclude that the element to which such sequences
> > converge is a real.
> >
> > The way you phrased it sounded presumptuous to me. Makes it sound
> > like Tony has to prove two things: each Cauchy sequence converges and
> > it converges to a real.
> >
> > Anyway, minor quibbles, I think. You're right whenever it comes to
> > things on which we agree, and wrong on the others. Same as everyone
> > else.
> >
> >
> I do like your James Harris quotes.
>
> Anyway, are you sure the set includes all rationals? I think this probably
> needs to be proven first, but am I being too hard on myself? It may be
> difficult to prove with this set. Is it not enough to prove that one can
> represent any n to within any finite accuracy epsilon with a finite
> representation? Why are the rationals involved, exactly?
.

Relevant Pages

  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... I don't know if there is a way of formally defining pi in terms of rational ... there are uncountably many reals and only ... using the rationals as a starting point. ... In cases where the Cauchy sequence (or Cantor sequence obeying the Cauchy ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > stevendaryl3016@xxxxxxxxx (Daryl McCullough) writes: ... >> for every Cauchy sequence of reals there is a Cauchy sequence ... >> of rationals converging to the same thing. ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... A real number can be defined by a Cauchy sequence ... rationals, which allows that there exists things such as the square ... I had a similar clarification a few posts back from Dave Marcus who observed that defining all the reals is different from defining one instance of the reals once the reals have been defined, and I need to read up on "defining the reals". ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... there is an ordered field with the LUB property. ... If every Cauchy sequence of rationals converges, then this limit exists and the sum exists. ... By construction of the reals, either through Dedekind cuts or, as I seem to recall, ...
    (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)