Re: Well Ordering the Reals

Daryl McCullough said:
> Tony Orlow says...
> >
> >Daryl McCullough said:
>
> >> >Listen, a countable number of bits, such that each is finite, is
> >> >supposedly sufficient to produce an uncountable number of bit
> >> >strings, ala 2^aleph_0, no?
> >>
> >> Yes, that's right. There are uncountably many infinite bitstrings.
> >Then my well ordering
>
> You don't have a well ordering of the reals.
>
> >only requires countably many bits, each in a finite
> >position, and therefore affording no infinite descending
> >sequence within the set of bitstrings. There is really
> >very little legitiamte doubt that this set is well ordered.
>
> If you are allowing infinite bitstrings, then you don't
> have a well ordering. There are infinite descending chains.
If there are countably many bits, then none is in an infinite location, is it?
> If you are only allowing finite bitstrings, then you do
> have a well ordering, but it doesn't include all the reals,
> only a dense subset.
What makes you say that? Cantor's putuative list of reals has countably many
bits, such that none is in a finite positions.
>
> Once again: Let B = the set of infinite bitstrings (indexed
> by finite natural numbers). Let FB = the set of finite bitstrings.
> Are you claiming to have a bijection between B and R, or are
> you claiming to have a bijection between FB and R. If you mean
> FB, then your bijection doesn't cover all the reals. If you mean
> B, then your set is not well-ordered. In either case, you are
> wrong to claim to have a well-ordering of the reals.
>From what you say, it sounds like any well ordering of the reals is impossible.
Is there something you know that Hilbert didn't?
>
> --
> Daryl McCullough
> Ithaca, NY
>
>

--
Smiles,

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

Relevant Pages

  • Re: Well Ordering the Reals
    ... There are uncountably many infinite bitstrings. ... You don't have a well ordering of the reals. ... Are you claiming to have a bijection between B and R, ... Daryl McCullough ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > Tony Orlow says... ... Your ordering of the reals is based on ... There is no well-ordering on the set of infinite bitstrings. ...
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  • Re: Well Ordering the Reals
    ... >> Tony Orlow says... ... If it is not an element of *N, then the fact that ...111 maps to the ... In the case of the standard mapping between infinite bitstrings ... Daryl McCullough ...
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  • Re: Well Ordering the Reals
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  • Re: Well Ordering the Reals
    ... >> mapping them to the set of bitstrings. ... >> on infinite bitstrings. ... The Dedekind and Cauchy constructions of the reals are PROVED ... It is all of it more than a little unclear to TO. ...
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