Re: Well Ordering the Reals


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.

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.

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.

- Randy

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... >>>TO's delusions, and has no existence in real mathematics, it is ... >> reals, when it actually just well-orders a countable subset. ... Show that for every non-empty set of your numbers which is bounded above ... there is a least upper bound in your set ...
    (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: 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)