Date: Jan 12, 2013 11:23 AM
Author: ross.finlayson@gmail.com
Subject: Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF<br> CANTOR: THE REALS ARE UNCOUNTABLE!
On Jan 11, 6:40 pm, Virgil <vir...@ligriv.com> wrote:

> In article

> <117f2274-de68-4a54-b90d-f3e423c3d...@c16g2000yqi.googlegroups.com>,

>

> WM <mueck...@rz.fh-augsburg.de> wrote:

> > On 11 Jan., 10:39, Virgil <vir...@ligriv.com> wrote:

> > > In article

> > > <3810bc42-c275-4897-94ba-8280508e9...@10g2000yqk.googlegroups.com>,

>

>

> > Correct. But Cantor's list requires decimals or equivalent

> > representations.

>

> Actually, Cantors original list for anti-diagonalization was of

> sequences of letters from {m,w}, not digits, and were not interpreted as

> numbers.

>

>

...

>

> Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben

> können.

>

> No one shall expel us from the Paradise that Cantor has created.

>

> David Hilbert

> --

"That's not "Cantor's original argument", for what he may have first

stated it.

http://en.wikipedia.org/wiki/Cantor's_theorem#History

For subsets M of N, the ordinal indices of S range from zero in alpha

through omega, let f_alpha(M) be onto {0} and f_omega(M) be onto {1},

then, G_alpha(M) = 1 - f_alpha(M) -> {1} = f_omega(M). Here f

satisfies the hypothesis of being a function from N at least into S

and doesn't see the contradiction. Here there are obviously infinite

ordinals between alpha and omega, between which there are functions

from N onto {0,1}. Basically this S has only one of the two values on

the ends, and two in the middle, with symmetry and reflection, and the

ordinal omega would look like 2^omega. Basically for each member of S

from zero, there is a corresponding bit-wise complement, in the same

order, in reverse, from omega, such that G_alpha = f_omega-alpha.

Thusly, G is not: not in S. Then, that would get back into Cantor

himself justifying counting backward from "limit ordinals", or that

omega is simply the next limit ordinal.

For Russell's, let phi-x be or include that "not-phi-x is false",

i.e., truth.

If all the propositions in the language have truth values, and the

theorem is about their self-referential content, then admit their self-

referential statement, here that in the language one, or the other, of

the statement, and its negation, is a statement in the language. "

Adam and Eve might have still been there hadn't they ate from the tree

of knowledge. Hilbert's Programme: completeness.

Regards,

Ross Finlayson