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Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF
CANTOR: THE REALS ARE UNCOUNTABLE!
Posted:
Jan 12, 2013 11:23 AM
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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
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