Date: Dec 28, 2012 2:09 AM Author: ross.finlayson@gmail.com Subject: Re: The Diagonal Argument 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.

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.

Powerset is order type is successor, Hancher, you rainbow-puking

regurgitist.

Ah, then excuse me, Hancher's quite monochromatic, besides

melodramatic, as "puke parrot", or for that matter, "puke maggot".

Yeah Hancher, Cantor discovered this construction some hundred years

ago, I'm glad you've dug back a few more years, try and catch up.

Regards,

Ross Finlayson