Re: Well Ordering the Reals

In article <MPG.1ddbfa626c13cef198a690@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> Daryl McCullough said:
> > Virgil says...
> >
> > >Since TO's alleged "apparent bijection" is merely an artifact of
> > >TO's delusions, and has no existence in real mathematics, it is
> > >certainly no big thing in mathematics.
> >
> > Actually, I think Tony is thinking (in his web page) of an
> > enumeration of a dense subset of the reals. That is, for any two
> > distinct reals r_1 and r_2 there is a real r in his set such that r
> > is between r_1 and r_2. He seems to think that this well-orders the
> > reals, when it actually just well-orders a countable subset.

> Well, that is what I am asking. How does one prove that such an
> ordered subset actually includes ALL the reals, rather than just the
> rationals or some other type of real? I don't think this enumeration
> misses one point on the line, but how do I prove this?

One method, based on Dedekind's construction, suggests itself:
Show that for every non-empty set of your numbers which is bounded above
(below) there is a least upper bound (greatest lower bound) in your set
of numbers.
.

Relevant Pages

  • Re: Well Ordering the Reals
    ... Tony Orlow wrote: ... >>>delusions, and has no existence in real mathematics, it is certainly no ... for any two distinct reals r_1 ... no problem accepting such things. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... Daryl McCullough said: ... >>delusions, and has no existence in real mathematics, it is certainly no ... for any two distinct reals r_1 ... > well-orders a countable subset. ...
    (sci.math)
  • Re: infinity
    ... range of an open interval in the reals. ... > LEAST UPPER BOUND of differences between values of members of that set, ... >>> I just said that I'm generating a new set by multiplying EVERY ... >> For sets of numbers mapped to the set of naturals, ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... compactification of the reals, as used in analysis. ... the surreals do not satisfy the least upper bound ... If the axiom of power set implys uncountable reals, ... The axiom of infinity guarantees that an infinite set exists. ...
    (sci.math)
  • Re: infinity
    ... > range of an open interval in the reals. ... >> LEAST UPPER BOUND of differences between values of members of that set, ... and finite lower bound,if any exists, need not abe greatest lower bound, ... >>> For sets of numbers mapped to the set of naturals, ...
    (sci.math)
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