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Topic: Cantor's Proofs
Replies: 57   Last Post: Nov 9, 2011 11:37 AM

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 William Hughes Posts: 1,555 Registered: 12/7/10
Re: Cantor's Proofs
Posted: Oct 15, 2011 2:34 AM

On Oct 15, 3:11 am, SPQR <S...@roman.gov> wrote:
> In article
>  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
>
> > > 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.

Date Subject Author
10/13/11 DBatchelo1
10/13/11 Ben Bacarisse
10/14/11 DBatchelo1
10/14/11 Tim Little
10/13/11 Ki Song
11/9/11 DBatchelo1
10/13/11 Peter Webb
10/14/11 DBatchelo1
10/14/11 Tim Little
10/14/11 William Hughes
10/14/11 Peter Webb
10/15/11 William Hughes
10/15/11 Peter Webb
10/15/11 alan.dennis.eaton@gmail.com
10/15/11 Peter Webb
10/15/11 SPQR
10/15/11 Graham Cooper
10/16/11 Patricia Shanahan
10/17/11 Barb Knox
10/18/11 Patricia Shanahan
10/18/11 Repeating Rifle
10/18/11 Graham Cooper
10/18/11 Patricia Shanahan
10/18/11 Graham Cooper
10/18/11 Repeating Rifle
10/18/11 Graham Cooper
10/22/11 two-jawed pliers
10/23/11 DBatchelo1
10/15/11 William Hughes
10/15/11 Peter Webb
10/15/11 alan.dennis.eaton@gmail.com
10/15/11 William Hughes
10/16/11 DBatchelo1
10/16/11 J. Antonio Perez M.
10/16/11 William Hughes
10/17/11 Graham Cooper
10/17/11 Peter Webb
10/17/11 Graham Cooper
10/17/11 Peter Webb
10/17/11 Graham Cooper
10/17/11 Peter Webb
10/17/11 Graham Cooper
10/17/11 Peter Webb
10/17/11 Graham Cooper
10/17/11 Peter Webb
10/17/11 |-| E R C
10/17/11 Peter Webb
10/17/11 |-| E R C
10/18/11 Tim Little
10/18/11 Graham Cooper
10/15/11 SPQR
10/15/11 William Hughes
10/15/11 Graham Cooper
10/15/11 SPQR
10/14/11 Tim Little
10/19/11 Daryl McCullough
10/19/11 Graham Cooper
10/22/11 Graham Cooper