
Re: The Diagonal Argument
Posted:
Dec 28, 2012 2:09 AM


On Dec 27, 9:35 pm, Virgil <vir...@ligriv.com> wrote: > In article > <c3b9462b682646fdbfe339c2d95ab...@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 twoletter 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 bitwise complement, in the same order, in reverse, from omega, such that G_alpha = f_omegaalpha. 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 phix be or include that "notphix is false", i.e., truth.
If all the propositions in the language have truth values, and the theorem is about their selfreferential 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 rainbowpuking 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

