Re: Well Ordering the Reals

Randy Poe said:
>
> Tony Orlow 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?
>
> You are asking how to prove a result which is known to be false. You
> can't do it within our axiomatic system.
Oh? Did someone just prove that a well ordering of the reals is impossible? Did
that happen over the weekend? Did Hilbert and Goedel know about this, and just
keep it s secret? What makes you think a well ordering is impossible?
>
> However, if you introduce a contradiction into the system, then you
> can prove all sorts of things. Any given theorem can be proven both
> true and false.
What contradiction (besides all the ones in there already)?
>
> In this case, a natural inconsistency to introduce would be the TO
> "axiom of getting to the end of unending sets", so that, for instance,
> it is declared by axiom that 1/3 is a member of the set {1/4, 1/4+1/16,
> 1/4+1/16+1/64, ...}
>
> So if you declare something like "all sequences contain their limits",
> then you can prove it pretty easily. Since that will give a result
> which is both provably true and provably false, it might cause
> other people problems. But you've shown in the past you have
> no problem accepting such things.
Of course I do, and this is exactly what I was saying about axiomatic systems.
You can prove anything just by declaring it as an axiom, but that's just a
waste of time. Axioms must be justified logically based on more fundamental
facts. But, you're not making much sense here.

>
> - Randy
>
>

--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

Relevant Pages

  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... > reals, so it can't possibly denumerate all of the uncountable reals. ... infinite bitstring to represent in that system, ... will require bit strings of infinite length. ...
    (sci.math)
  • Re: An uncountable countable set
    ... Tony Orlow wrote: ... And what is the smallest finite distance? ... That does not match anyone else's set of reals. ... LUB has a LUB which is not a member of the sequence. ...
    (sci.math)
  • Re: An uncountable countable set
    ... Tony Orlow wrote: ... David R Tribble wrote: ... then we'll talk about "covering the reals". ... formal proof of the equivalence between the H-riffics and the reals. ...
    (sci.math)
  • Re: Extending the reals
    ... Tony Orlow wrote: ... reals and 0, then the reals consist of three disjoint sets, ... a member of H_0, and vice versa. ... Except that the negative suprareals in H_1 are not connected ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > David R Tribble said: ... > Tony Orlow wrote: ... The first 20 reals ...
    (sci.math)