In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 18 Mrz., 00:59, Virgil <vir...@ligriv.com> wrote: > > > > > Why does WM claim that after what WM calls "Cantor's list" has been > > diagonalized, he can include all anti-diagonals, when it is always > > possible to find others that have been so far overlooked? > > This simply and exactly shows that it is inconsistent to assume one of > both powers to be stronger. > > Every list yields another diagonal. > And every diagonal can be included in another list.
The issue is whether a list can exist for which a non-member of that list can not exist, and Wm concedes that it cannot. Thus WM concedes that Canntors diagonal argument is valid. > > Both these facts are the two sides of infinity. Cantor did that false > step to choose one as more justified than the other.
That only one of them is reuired to prove wha Cantor claimsed seems t have evaded WM's comprehension.
But then so much else does as well,
> The isomorphism is from |R,+,* to |R,+,*. Only in one case the > elements of |R are written as binary sequences and the other time as > paths of the Binary Tree. Virgil is simply too stupid to understand > that.everal flaws in WM's claim that the identity map on induces a linear map on 2^|N.
WM's flaws in making that claim work include, but are not necessarily limited to:
(1) not all members of |R will have any such binary expansions, only those between 0 and 1, so that not all sums of vectors will "add up" to be vectors within his alleged linear space, and
(2) some reals (the positive binary rationals strictly between 0 and 1) will have two distinct and unequal-as-vectors representations, requiring that some real numbers not be equal to themselves as a vectors, and
(3) WM's method does not provide for the negatives of any of the vectors that he can form.
On the basis of the above problems, and possibly others as well that I have not yet even thought of, I challenge WM's claim to have represented the set |R as the set of all binary sequences, much less to have imbued that set of all binary sequences with the structure of a real vector space or the showed the identity mapping to be a linear mapping on his set of "vectors". --