```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 firststated it.http://en.wikipedia.org/wiki/Cantor's_theorem#HistoryFor subsets M of N, the ordinal indices of S range from zero in alphathrough 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 leastinto S and doesn't see the contradiction.  Here there are obviouslyinfinite ordinals between alpha and omega, between which there arefunctions from N onto {0,1}.  Basically this S has only one of the twovalues on the ends, and two in the middle, with symmetry andreflection, and the ordinal omega would look like 2^omega.  Basicallyfor each member of S from zero, there is a corresponding bit-wisecomplement, in the same order, in reverse, from omega, such thatG_alpha = f_omega-alpha.  Thusly, G is not: not in S.Then, that would get back into Cantor himself justifying countingbackward from "limit ordinals", or that omega is simply the next limitordinal.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 thetheorem is about their self-referential content, then admit their self-referential statement, here that in the language one, or the other, ofthe statement, and its negation, is a statement in the language.Powerset is order type is successor, Hancher, you rainbow-pukingregurgitist.Ah, then excuse me, Hancher's quite monochromatic, besidesmelodramatic, as "puke parrot", or for that matter, "puke maggot".Yeah Hancher, Cantor discovered this construction some hundred yearsago, I'm glad you've dug back a few more years, try and catch up.Regards,Ross Finlayson
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