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Virgil
Posts:
4,483
Registered:
1/6/11
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Re: Cantor's absurdity, once again, why not?
Posted:
Mar 19, 2013 2:34 PM
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In article
<bce9fd0e-a1b4-4157-af24-cb22d80057cc@c10g2000vbt.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 17 Mrz., 07:11, fom <fomJ...@nyms.net> wrote:
> > On 3/16/2013 10:55 AM, WM wrote:
> >
> > > On 16 Mrz., 16:01, fom <fomJ...@nyms.net> wrote:
> >
> > >> perhaps you could explain what you mean
> > >> by "given object" and how an immaterial
> > >> object can be given.
> >
> > > It cannot be given other than by naming it (except from clumsy
> > > approaches by means of sign language). How to name some numbers, and
> > > rules how to invent further names, that can be understood by others,
> > > who were taught the same rules, is taught in school, university and
> > > other sources.
> >
> > What then are some examples
> > of rules that invent these
> > further names?
>
> If 5 and 6 are given, mathematics defines how to produce 11.
Not until decimal notation, at least up to 11, is also given.
> >
> > The point of this question is
> > that you claim such rules but
> > ignore the work of others who
> > have steadfastly worked at clarifying
> > the nature of such rules as a
> > matter of scientific principle
> > (in the wider epistemological
> > sense).
>
> I do not ignore these rules, but in some instances I can show that
> they are contradictory.
WM often claims to be able to show things, but the only thong he shows
with any reliability is his incompetence as a mathematician.
> 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".
=========================================================================
WM has several times claimed that the standard bijection from the set of
all binary sequences to the set of all paths of a Complete Infinite
Binary Tree is a linear mapping from the set of all binary sequences
regarded as a linear space over |R to the set of all paths of a CIBT.
While the obvious mapping is easily shown to be bijective, it fails to
be a linear mapping as WM describes it:
> 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 of binaries, 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 so
that two such pairs of binary sequences can only map to a single real
thus also only to a single path, so that the mapping cannot be a
bijection, and
(3) WM's method does not provide for the negatives of any of the vectors
that he can form, so his "space" does not qualify as a linear space in
that way, either.
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 linearly
injected the set of binary sequences into the set |R or the image of the
set of binary sequnces in |R linearly ONTO the set of all paths.
Note that while WM's model doe not achieve what he claims for it, there
is another model, which a reasonably competent mathematician should be
able to find, which does make the mapping into an isomorphism of linear
spaces. I will produce this model when WM concedes his error, or at
least no longer claims it is not an error.
It is a shame that someone so obviously of limited ability at
mathematics as WM should feel himself so driven to try and correct his
betters.
>
> Regards, WM
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