Re: Well Ordering the Reals


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.

Declaring it to be false does not declare well-ordering to be
impossible.

- Randy

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... >> Randy Poe said: ... > Declaring it to be false does not declare well-ordering to be ... Prev by Date: ...
    (sci.math)
  • Re: infinity
    ... > Randy Poe said: ... >> Tony Orlow wrote: ... >>> I never said the range had to be an element of any arbitrary set. ... the "value range" of a set of reals only ...
    (sci.math)
  • 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. ...
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  • 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: infinity
    ... > Tony Orlow wrote: ... >> Randy Poe said: ... I'm not talking for the moment about this notation. ... member of the set of naturals starting at 0. ...
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