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Re: Cantor's Proofs
Posted:
Oct 15, 2011 2:34 AM
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On Oct 15, 3:11 am, SPQR <S...@roman.gov> wrote:
> In article
> <d5b61c9e-257a-4ba8-b91c-bfdb29ece...@u2g2000yqc.googlegroups.com>,
> William Hughes <wpihug...@gmail.com> wrote:
>
>
>
> > On Oct 15, 12:46 am, "Peter Webb"
> > <webbfam...@optusnetDIESPAMDIE.com.au> wrote:
> > > "William Hughes" <wpihug...@gmail.com> wrote in message
>
> > >news:76d8ba46-e5f8-47b9-8b48-af478fedc38a@f5g2000vbz.googlegroups.com...
> > > On Oct 13, 10:02 pm, "Peter Webb"
>
> > > <webbfam...@optusnetDIESPAMDIE.com.au> wrote:
> > > > The standard presentation of Cantor's second proof shows that any given
> > > > (purported) list of all Reals must contain a Real not on the list.
>
> > > > Therefore Cantor's proof shows that it is impossible to explicitly form a
> > > > list of all Reals
>
> > > Where did you get "explicitly" and what does it mean?
> > > The proof applies to any list.
>
> > > ______________________________
> > > Explicitly means you provide every Real on the list.
>
> > > > As you point out, this is not necessarily the same as being uncountable.
> > > > There are countable sets (eg all computable numbers) that are also
> > > > impossible to explicitly list.
>
> > > There is certainly a list of the computable numbers, however there is
> > > no
> > > computable list. Does "explicit list" mean "computable list"?
>
> > > - William Hughes
>
> > > _____________________________________________
> > > Yes
>
> > Cantor's proof is not restricted to computable lists.
> > You need to drop the qualifier "explicitly"
>
> > > Cantor's proof is constructive. It starts with a purported list of all
> > > Reals, then invokes an algorithm based upon knowledge of the nth digit of
> > > the nth entry to form a number not on the list. This proves that there can
> > > be no list of all Reals. That is not quite the same thing as being
> > > uncountable.
>
> > By definition of uncountable it is exactly the same thing.
>
> > - William Hughes
>
> At least given that there are injections from N to R it is the same
> thing.
I am unaware of any theory in which (a set isomorphic to) N is not a
subset of
R. (Indeed, I fail to see how you could have N not a subset of R)
In any case, uncountable means unlistable, so it is the same thing.
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