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Topic: The Diagonal Argument
Replies: 28   Last Post: Dec 29, 2012 12:11 AM

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Virgil

Posts: 7,011
Registered: 1/6/11
Re: The Diagonal Argument
Posted: Dec 28, 2012 4:27 AM
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In article
<b75568d4-cb63-495d-a6fa-4189b90eacfe@s6g2000pby.googlegroups.com>,
"Ross A. Finlayson" <ross.finlayson@gmail.com> wrote:

> On Dec 27, 9:35 pm, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <c3b9462b-6826-46fd-bfe3-39c2d95ab...@pe9g2000pbc.googlegroups.com>,
> >  Graham Cooper <grahamcoop...@gmail.com> wrote:
> >

> > > one must consider the audience Virgil!
> >
> > > SWAPPING DIGITS DOWN THE DIAGONAL
> >
> > > seems to be the only mathematics he can grasp!
> >
> > Actually, Cantor's original argument does not even use digits.
> >
> > Cantor considers the set, S, of functions from the set of naturals |N as
> > domain, to the two-letter set of letters {m,w}, and shows that there
> > cannot be any surjective mapping f: |N -> S by  constructing a member g
> > of S not in Image(f)
> >
> > Since  f: |N -> S, each f(n) is a function from |N to {m,w}
> > So that when  g(n) is a member of {m,w}\f(n)(n) for each n, then g is
> > not a member of S.
> > --

>
> That's not "Cantor's original argument", for what he may have first
> stated it.


If is in a considerably different form, but is precisely the idea of
Cantor's 'diagonal' argument, based on the set of all infinite sequences
of letters taken from {m,w}.

Note that Cantor had a fair number of other theorems re infiniteness
other than the one called his diagonal argument.
--





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