Re: Well Ordering the Reals

Randy Poe said:
>
> Tony Orlow wrote:
> > 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?
>
> You are asking how to prove that the set for which you are
> constructing a well-ordering is R.
>
> This is a result which is false.
Prove it, please.
>
> Declaring it to be false does not declare well-ordering to be
> impossible.
>
> - Randy
>
>

--
Smiles,

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

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