
Re: The Diagonal Argument
Posted:
Dec 28, 2012 12:16 PM


On Dec 28, 1:27 am, Virgil <vir...@ligriv.com> wrote: > In article > <b75568d4cb63495da6fa4189b90ea...@s6g2000pby.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > > > > > > > > > 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. > > 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 it seems clear that you do actually read the attempts of others to develop frameworks and structures of what would be developments in mathematics, but it is as well clear that you definitely have a penchant for tearing down said arguments without building them up. Then, while here your usual histrionics haven't yet erupted: on to the developments above.
Here, then it was presented that a reasonably simple construction of set S, of functions f: N > {0,1}, sees that f_alpha(x) = 1  f_omega alpha(x), and that G_alpha = f_omegaalpha, with the hypothesis satisfied and contradiction not following, thus a difference in result. (And, that's not much of a "diagonal" argument except insofar as iteratively building for each element of an enumeration with an infinite enumeration of its structure, a differing element. Here, transfinite ordinals have the first omegamany elements of f having complements symmetrically from the end.)
Basically then this sees establishing a symmetry, between zero, and the 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, that such a thing is reasonable. Graham, to disprove a proof by contradiction, 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 was described that a language with structure only having true propositions, would not see the result follow, for example of constructive 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 the constructive nature of extremes, in that our simple foundations must see comprehension of all our constructions, as above is a development for seeing that Cantor's indicator function theorem doesn't necessarily hold, and that Russell's correlatenegating theorem doesn't necessarily hold, then for someone interested in seeing countable reals, there would be various development for Cantor's nested intervals, and Cantor's antidiagonal, and Cantor's powerset results.
Then, for the general notion of the antidiagonal argument or diagonalization and the diagonal method, what I've seen is that generally in the extreme and the infinite, there is establishment of a symmetry principle, that then the diagonal is flattened. This then would be a general consideration of diagonalization, and here, squaring.
Regards,
Ross Finlayson

