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

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ross.finlayson@gmail.com

Posts: 1,218
Registered: 2/15/09
Re: The Diagonal Argument
Posted: Dec 28, 2012 12:16 PM
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On Dec 28, 1:27 am, Virgil <vir...@ligriv.com> wrote:
> In article
> <b75568d4-cb63-495d-a6fa-4189b90ea...@s6g2000pby.googlegroups.com>,
>  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
>
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>
>

> > 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.
> --


Hancher, the "puke parrot" bit is largely for comedic effect, yes it
seems clear that you do actually read the attempts of others to
develop frameworks and structures of what would be developments in
mathematics, but it is as well clear that you definitely have a
penchant for tearing down said arguments without building them up.
Then, while here your usual histrionics haven't yet erupted: on to
the developments above.

Here, then it was presented that a reasonably simple construction of
set S, of functions f: N -> {0,1}, sees that f_alpha(x) = 1 - f_omega-
alpha(x), and that G_alpha = f_omega-alpha, with the hypothesis
satisfied and contradiction not following, thus a difference in
result. (And, that's not much of a "diagonal" argument except insofar
as iteratively building for each element of an enumeration with an
infinite enumeration of its structure, a differing element. Here,
transfinite ordinals have the first omega-many elements of f having
complements symmetrically from the end.)

Basically then this sees establishing a symmetry, between zero, and
the first limit ordinal, in a structure then of 0->w and w->0. Now,
this is an example of one of the many ideas put forth by Cantor, that
such a thing is reasonable. Graham, to disprove a proof by
contradiction, it's one thing to show that the result doesn't follow,
another to show the hypothesis is satisfied.

As well in reference to Russell's correlate result, there was
described that a language with structure only having true
propositions, would not see the result follow, for example of
constructive results of a closed language in a consistent universe,
that there was an untrue one.

Then for the reader interested in roots of foundations and as well the
constructive nature of extremes, in that our simple foundations must
see comprehension of all our constructions, as above is a development
for seeing that Cantor's indicator function theorem doesn't
necessarily hold, and that Russell's correlate-negating theorem
doesn't necessarily hold, then for someone interested in seeing
countable reals, there would be various development for Cantor's
nested intervals, and Cantor's antidiagonal, and Cantor's powerset
results.

Then, for the general notion of the antidiagonal argument or
diagonalization and the diagonal method, what I've seen is that
generally in the extreme and the infinite, there is establishment of a
symmetry principle, that then the diagonal is flattened. This then
would be a general consideration of diagonalization, and here,
squaring.

Regards,

Ross Finlayson



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