```Date: Dec 28, 2012 12:16 PM
Author: ross.finlayson@gmail.com
Subject: Re: The Diagonal Argument

On Dec 28, 1:27 am, Virgil <vir...@ligriv.com> wrote:> In article> <b75568d4-cb63-495d-a6fa-4189b90ea...@s6g2000pby.googlegroups.com>,>  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:>>>>>>>>>> > 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.>> If is in a considerably different form, but is precisely the idea of> Cantor's 'diagonal' argument, based on the set of all infinite sequences> of letters taken from {m,w}.>> Note that Cantor had a fair number of other theorems re infiniteness> other than the one called his diagonal argument.> --Hancher, the "puke parrot" bit is largely for comedic effect, yes itseems clear that you do actually read the attempts of others todevelop frameworks and structures of what would be developments inmathematics, but it is as well clear that you definitely have apenchant for tearing down said arguments without building them up.Then, while here your usual histrionics haven't yet erupted:  on tothe developments above.Here, then it was presented that a reasonably simple construction ofset S, of functions f: N -> {0,1}, sees that f_alpha(x) = 1 - f_omega-alpha(x), and that G_alpha = f_omega-alpha, with the hypothesissatisfied and contradiction not following, thus a difference inresult.  (And, that's not much of a "diagonal" argument except insofaras iteratively building for each element of an enumeration with aninfinite enumeration of its structure, a differing element.  Here,transfinite ordinals have the first omega-many elements of f havingcomplements symmetrically from the end.)Basically then this sees establishing a symmetry, between zero, andthe first limit ordinal, in a structure then of 0->w and w->0.  Now,this is an example of one of the many ideas put forth by Cantor, thatsuch a thing is reasonable.  Graham, to disprove a proof bycontradiction, it's one thing to show that the result doesn't follow,another to show the hypothesis is satisfied.As well in reference to Russell's correlate result, there wasdescribed that a language with structure only having truepropositions, would not see the result follow, for example ofconstructive results of a closed language in a consistent universe,that there was an untrue one.Then for the reader interested in roots of foundations and as well theconstructive nature of extremes, in that our simple foundations mustsee comprehension of all our constructions, as above is a developmentfor seeing that Cantor's indicator function theorem doesn'tnecessarily hold, and that Russell's correlate-negating theoremdoesn't necessarily hold, then for someone interested in seeingcountable reals, there would be various development for Cantor'snested intervals, and Cantor's antidiagonal, and Cantor's powersetresults.Then, for the general notion of the antidiagonal argument ordiagonalization and the diagonal method, what I've seen is thatgenerally in the extreme and the infinite, there is establishment of asymmetry principle, that then the diagonal is flattened.  This thenwould be a general consideration of diagonalization, and here,squaring.Regards,Ross Finlayson
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