Virgil
Posts:
8,833
Registered:
1/6/11


Re: The Diagonal Argument
Posted:
Dec 28, 2012 4:27 AM


In article <b75568d4cb63495da6fa4189b90eacfe@s6g2000pby.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson@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. 

