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


Re: Matheology � 263
Posted:
May 17, 2013 2:56 AM


In article <908dc96904a440d7acdd2b0ac57911c0@a10g2000pbr.googlegroups.com>, Graham Cooper <grahamcooper7@gmail.com> wrote:
> On May 17, 4:08 pm, Virgil <vir...@ligriv.com> wrote: > > In article > > <9d6b39cdb3494d9ba34ea759e81cb...@ul7g2000pbc.googlegroups.com>, > > Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > > > On May 17, 11:51 am, Virgil <vir...@ligriv.com> wrote: > > > > In article > > > > <3fb4082ca45547a8a931c219fd2fb...@qc10g2000pbb.googlegroups.com>, > > > > Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > > > On May 17, 10:57 am, Virgil <vir...@ligriv.com> wrote: > > > > > > > OK, use > > > > > > > > > f(1) = { 1 , 3 , 6 } > > > > > > > f(2) = { 1 , 5 , 11 } > > > > > > > f(3) = { 2 , 4 , 6, 8 , 10 , ... } > > > > > > > f(4) = { 4 , 5, 6, 7, 8 } > > > > > > > > > f(n) = N  n>4 > > > > > > > > > What is set S? > > > > > > > > Note that there are only 5 subsets of N that ARE in the image of > > > > > > your > > > > > > 'f' out of uncountably many subsets of N to chose from that will > > > > > > be > > > > > > S's, i.e., subset of N but not values of f. > > > > > > > > They are > > > > > > every finite subset of N having > > > > > > less that 3 members > > > > > > or exactly 4 members, > > > > > > or more than 5 members, > > > > > > or containing either 2, or a natural larger than 8, > > > > > > and every infinite subset of N other than f(3) and N. > > > > > > > > Take your pick. > > > > > > ` > > > > > So Cantor's Method does not work? > > > > Since there are far more sets not in the image of any function from N > > to 2^N than in its image, why are you so hot to get one in particular? > > > > > > > > > Your method (white box inspection) does not work for every possible f, > > > as such it is of no consequence. > > > > Name one it does not work for! > > > > > > > > > GIVEN A SPECIFIC f, WHAT IS CANTORS MISSING SET? > > > > If you could pick a subset of N at random, it would almost certainly > > not be in the image of any randomly chosen function. > > > > > > > > > All your answers are wrong! > > > > They are not in the image of the function you presented, which is all > > they are asked to be. > > > > NO! THIS is the question. > > f(1) = { 1 , 3 , 6 } > f(2) = { 1 , 5 , 11 } > f(3) = { 2 , 4 , 6, 8 , 10 , ... } > f(4) = { 4 , 5, 6, 7, 8 } > > f(n) = N  n>4 > > What is set S? > > from 1 function f() you get 1 set S
Actually, one has far more sets, S, not in the image of any such function than are in its image.
The smallest is {}, every one element set also works, as does every two element set, and all but two of the countably many three element sets. > > You can post *piffle* AFTER you've > > CALCULATED_CANTORS_MISSING_SET
Cantors does not say that there is no more than one such set, he just said that there is at least one, and I have provided at least one, in fact uncountably many of them. 

