Date: Mar 19, 2013 2:30 PM
Author: Virgil
Subject: Re: Matheology � 224

In article 
WM <> wrote:

> On 19 Mrz., 00:44, William Hughes <> wrote:
> [Topic deleted that is irrelevant for most readers.]

> > > But the more pressing question is: You construct a list such that
> > > every line contains all preceding contents. You get ready, i.e., the
> > > list contains all that it can contain. Nevertheless there is no line
> > > that contains everything that the list contains.

> >
> > Yep, no last line.

> Do you recognize that this is no explanation for your assertion that
> in our list more than one line contain more than one line contains?

Infinitely many finite lines, or finite sets of any nature, each
strictly larger than its predcessors, will contain more than any one
line, or any finite set of those lines or sets.
> Do you accept that a set of infinitely many lines contains at least
> one subset of two lines?

It contains infinitely many such subsets.
> Do you accept that every static (= existing in the sense of set
> theory) set of lines has a (fixed and knowable) first line?

That depends on how the lines in such sets are ordered. If those sets of
lines are all well-ordered sets of lines, then yes.

Otherwise, not necessarily.


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