Re: Well Ordering the Reals

imaginatorium@xxxxxxxxxxxxx said:
>
> Tony Orlow wrote:
> > Daryl McCullough said:
>
> > > 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?
>
> Another exercise for you, Tony. Describe, preferably in your own words,
> the difference between the standard meanings of "enumeration (indexed
> by the naturals)" and "well-ordering".
>
> Brian Chandler
> http://imaginatorium.org
>
>
An enumeration is simply a listing, although usually with some ordering. A well
ordering appears to be an ordering such that every element is comparable with
another in a linear order, and every subset has a smallest element. This second
criterion seems to be at odds with the explicit well ordering of any
uncountable set as far as I can tell. I am sure I have not stated all the
conditions of a well ordering, and am ready for correction.
--
Smiles,

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

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