Re: Matheology § 224

[William Hughes]

On Mar 14, 10:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:

<snip>

>... consider the list of finite initial segments of natural numbers
>
> 1
> 1, 2
> 1, 2, 3
> ...
>
> According to set theory it contains all aleph_0 natural numbers in its
> lines. But is does not contain a line containing all natural numbers.
> Therefore it must be claimed that more than one line is required to
> contain all natural numbers. This means at least two line are
> necessary. There are no special lines necessary, but there must be at
> least two. In this case, however, we can prove, by the construction of
> the list, that every union of a pair of lines is contained in one of
> the lines. This contradicts the assertion that all natural numbers
> exist and are in lines of the list.


Nope.

Nope, two lines are necessary but not sufficient.

Two lines can never do a better job than 1.

Any finite number of lines is necessary but not sufficient.

Any finite number of lines can never do a better job than 1.

An infinite number of lines is necessary and sufficient

An infinite number of lines can do a better job than 1

[In potential infinity things go

Any number of findable lines is not sufficient

An unfindable line is necessary and sufficient

An unfindable line can do a better job than
any number of findable lines.
]

[WM]

On 15 Mrz., 17:13, William Hughes <wpihug...@gmail.com> wrote:
> On Mar 14, 10:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> <snip>
>
>
>
>
>
> >... consider the list of finite initial segments of natural numbers
>
> > 1
> > 1, 2
> > 1, 2, 3
> > ...
>
> > According to set theory it contains all aleph_0 natural numbers in its
> > lines. But is does not contain a line containing all natural numbers.
> > Therefore it must be claimed that more than one line is required to
> > contain all natural numbers. This means at least two line are
> > necessary. There are no special lines necessary, but there must be at
> > least two. In this case, however, we can prove, by the construction of
> > the list, that every union of a pair of lines is contained in one of
> > the lines. This contradicts the assertion that all natural numbers
> > exist and are in lines of the list.
>
> Nope.
>
> Nope, two lines are necessary but not sufficient.

Let's first prove that already two cannot be necessary by the fact
that two always can be replaced by one of them without changing the
contents. Then it is clear that two or more cannot be necessary and
from this immediately follows that they also cannot be sufficient.

>
> Two lines can never do a better job than 1.
>
> Any finite number of lines is necessary but not sufficient.

Wrong. Why do you resist to apply logic?

>
> Any finite number of lines can never do a better job than 1.
>
> An infinite number of lines is necessary and sufficient

Exercise: If of any two line one is not necessary, how many of
infinitely many lines are not necessary?

>
> An infinite number of lines can do a better job than 1

That is a confession of irrational belief. With exactly the same right
you could state: An infinite number of even naturals contains an odd
natural.

You may claim so, but it is not part of mathematics. You should
accept: If someone claims infinity and is not even able to show two,
then his claim is nothing that could be provable, hence nothing that
belongs to mathematics.

>
> [In potential infinity things go
>
> Any number of findable lines is not sufficient
>
> An unfindable line is necessary and sufficient

In any case the last line contains every number of the list. This is
so by construction. We have the choice between 1 line (in potential
infinity) and 0 lines (in actual infinity).

Regards, WM

[Virgil]

In article
<37f8b921-a2db-46d0...@k14g2000vbv.googlegroups.com>,

Or it would if you could find it.
--

[William Hughes]

On Mar 15, 8:34 pm, WM <mueck...@rz.fh-augsburg.de> wrote:


> Let's first prove that already two cannot be necessary by the fact
> that two always can be replaced by one of them without changing the
> contents.

This is true but the fact that the two lines are
necessary has nothing to do with their contents. Two lines
cannot be replaced by one of them without changing the number
of lines.

Consider the case is potential infinity.
A set of lines, K, that has an unfindable last number
must contain at least two findable lines.
The fact that these two lines are necessary has
nothing to do with the contents of the lines.

[Virgil]

In article
<f5511d40-6a50-4a43...@ia3g2000vbb.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 15 Mrz., 17:13, William Hughes <wpihug...@gmail.com> wrote:
> > On Mar 14, 10:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > <snip>
> >
> >
> >
> >
> >
> > >... consider the list of finite initial segments of natural numbers
> >
> > > 1
> > > 1, 2
> > > 1, 2, 3
> > > ...
> >
> > > According to set theory it contains all aleph_0 natural numbers in its
> > > lines. But is does not contain a line containing all natural numbers.
> > > Therefore it must be claimed that more than one line is required to
> > > contain all natural numbers. This means at least two line are
> > > necessary. There are no special lines necessary, but there must be at
> > > least two. In this case, however, we can prove, by the construction of
> > > the list, that every union of a pair of lines is contained in one of
> > > the lines. This contradicts the assertion that all natural numbers
> > > exist and are in lines of the list.
> >
> > Nope.
> >
> > Nope, two lines are necessary but not sufficient.
>
> Let's first prove that already two cannot be necessary by the fact
> that two always can be replaced by one of them without changing the
> contents. Then it is clear that two or more cannot be necessary and
> from this immediately follows that they also cannot be sufficient.

Enough more than two line can be necessary and can be sufficient,
outside of WMytheology. Inside WMytheology it is not apparent that
anything can be either necessary or sufficient, since there cannot in
WMytheology be all lines, or even infinitely many.

> >
> > Two lines can never do a better job than 1.
> >
> > Any finite number of lines is necessary but not sufficient.
>
> Wrong. Why do you resist to apply logic?

A positive finite number of lines in necessary unless no set of lines
can be sufficient. But the set of all lines (at least outside
WMytheology) is certainly sufficient.

> >
> > Any finite number of lines can never do a better job than 1.
> >
> > An infinite number of lines is necessary and sufficient
>
> Exercise: If of any two line one is not necessary, how many of
> infinitely many lines are not necessary?

Infinitely many are not necessary provided infinitely many are still
used.

For example, for any prime, the set of infinitely many lines ending in
multiples of that prime are sufficient.
Or even the set of lines ending in a prime.
Or in the square o a prime,
or the cube of a prime,
and so on ad infinitum.

> >
> > An infinite number of lines can do a better job than 1
>
> That is a confession of irrational belief. With exactly the same right
> you could state: An infinite number of even naturals contains an odd
> natural.

Only in WMytheology could an infinite number of even naturals contain an
odd natural as life in WMytheology is totally unnatural..

>
> You may claim so, but it is not part of mathematics.

No one outside of WMytheology is claiming so.

You should
> accept: If someone claims infinity and is not even able to show two

A finite number of lines/FISs is not enough,
at least outside of WMytheology
because outside of WMytheology for any finite number of lines there is a
natural in their union whose successor is not in their union.
Wm claims tha this cannot happen in his WMytheology!

> >
> > [In potential infinity things go
> >
> > Any number of findable lines is not sufficient
> >
> > An unfindable line is necessary and sufficient
>
> In any case the last line contains every number of the list.

That presumes a last line, but outside WMytheology in order to even be a
line at all, a set of naturals has to have a successor set.

> This is
> so by construction.

Then why is WM still so unable to construct the set of all binary
sequences and the set of paths of a CIBT in the form of linear spaces
and the obvious bijection between them as a linear mapping?

We have the choice between 1 line (in potential
> infinity) and 0 lines (in actual infinity).

Is that a Royal "We"?

If so, it is being badly misused.
--

[Virgil]

In article
<0c0b10c7-65f5-4de7...@r8g2000vbj.googlegroups.com>,

How can a set of lines be made to HAVE an unfindable last line unless
one can somehow FIND that allegedly unfindable last line?

Outside of WMytheology, it can't.

Which demonstrates the MYTHeology in
W-MYTH-eology
--

[WM]

On 15 Mrz., 23:27, William Hughes <wpihug...@gmail.com> wrote:
> On Mar 15, 8:34 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > Let's first prove that already two cannot be necessary by the fact
> > that two always can be replaced by one of them without changing the
> > contents.
>
> This is true but the fact that the two lines are
> necessary has nothing to do with their contents.  Two lines
> cannot be replaced by one of them without changing the number
> of lines.

Why should line-numbers be changed? Perhaps we are misunderstanding
each other. This is my claim:

Here is a list with three lines containing five natural numbers

1) 1, 2, 3, 4
2) 1, 2, 3, 4, 5
3) 1, 2, 3, 4

We can remove lines 1 and 3 without reducing the contents ofthe list.
Line number 2 remains line number 2.

> Consider the case is potential infinity.
> A set of lines, K, that has an unfindable last number
> must contain at least two findable lines.
> The fact that these two lines are necessary has
> nothing to do with the contents of the lines.

Here I would like to see an example.

But I would ask you in advance: Do you agree that every non-empty set
of natural numbers (including line-numbers) has a smallest element? Or
do you believe that here the exception proves the rule?

Regards, WM

[WM]

On 16 Mrz., 00:39, Virgil <vir...@ligriv.com> wrote:

> > Let's first prove that already two cannot be necessary by the fact
> > that two always can be replaced by one of them without changing the
> > contents. Then it is clear that two or more cannot be necessary and
> > from this immediately follows that they also cannot be sufficient.
>
> Enough more than two line can be necessary and can be sufficient,

Do you agree that every non-empty set of line-numbers contains a least
element?

> > Wrong. Why do you resist to apply logic?
>
> A positive finite number of lines in necessary

A positive finite number of lines contains a least element.

> We have the choice between 1 line (in potential
>
> > infinity) and 0 lines (in actual infinity).
>
> Is that a Royal "We"?

No it includes everybody, many don't know though.

Regards, WM

[WM]

On 16 Mrz., 01:02, Virgil <vir...@ligriv.com> wrote:

> How can a set of lines be made to HAVE an unfindable last line

Simply count.

> unless
> one can somehow FIND that allegedly unfindable last line?

Simply continue counting.

Regards, WM

[William Hughes]

On Mar 16, 10:30 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 15 Mrz., 23:27, William Hughes <wpihug...@gmail.com> wrote:
>
> > On Mar 15, 8:34 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > Let's first prove that already two cannot be necessary by the fact
> > > that two always can be replaced by one of them without changing the
> > > contents.
>
> > This is true but the fact that the two lines are
> > necessary has nothing to do with their contents.  Two lines
> > cannot be replaced by one of them without changing the number
> > of lines.
>
> Why should line-numbers be changed? Perhaps we are misunderstanding
> each other.

I said "number of lines" not "line-numbers".
If you replace two lines by one of them, you do
not change the contents, but you do change the number
of lines. Since the number of lines is important
and the contents are not, you cannot replace
two lines with one line.

[WM]

Ok, I understand. Anyhow, if the number of lines is not empty, then
there must remain at least one line as a necessary line. That line has
a line-number in the original list. I think that if one more more
lines are necessary, as you claim, then there must be a set of line-
numbers which is not empty and, therefore, has a least element.

Do you think that this is unmathematical?

Regards, WM

[William Hughes]

On Mar 16, 1:16 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 16 Mrz., 12:26, William Hughes <wpihug...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On Mar 16, 10:30 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > On 15 Mrz., 23:27, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > On Mar 15, 8:34 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > > Let's first prove that already two cannot be necessary by the fact
> > > > > that two always can be replaced by one of them without changing the
> > > > > contents.
>
> > > > This is true but the fact that the two lines are
> > > > necessary has nothing to do with their contents.  Two lines
> > > > cannot be replaced by one of them without changing the number
> > > > of lines.
>
> > > Why should line-numbers be changed? Perhaps we are misunderstanding
> > > each other.
>
> > I said "number of lines" not "line-numbers".
> > If you replace two lines by one of them, you do
> > not change the contents, but you do change the number
> > of lines.  Since the number of lines is important
> > and the contents are not, you cannot replace
> > two lines with one line.
>
> Ok, I understand. Anyhow, if the number of lines is not empty, then
> there must remain at least one line as a necessary line.

Not a particular line. This is similar to
the case where any set of lines with an unfindable
last line has at least one "necessary" findable line.
This line has a line number in the original
list but we can choose the "necessary"
findable line to have any line number we want.
The fact that more than one findable line
is "necessary" does not mean there must
be a set of line numbers which is nonempty
and has a least element.

[WM]

On 16 Mrz., 18:10, William Hughes <wpihug...@gmail.com> wrote:

> > Ok, I understand. Anyhow, if the number of lines is not empty, then
> > there must remain at least one line as a necessary line.
>
> Not a particular line.  This is similar to
> the case where any set of lines with an unfindable
> last line has at least one "necessary" findable line.
> This line has a line number in the original
> list but we can choose the  "necessary"
> findable line to have any line number we want.

No, it is always the last line. We call it unfindable or unfixable
because as soon as we have found it, it is no longer the last line.

> The fact that more than one findable line
> is "necessary" does not mean there must
> be a set of line numbers which is nonempty

> and has a least element.-

That is interesting. We have a set of natural numbers, so called line-
numbers of necessary findable lines. This fact does not mean that the
set of so called line-numbers of necessary findable lines is nonempty

and has a least element.

I understand that an empty set need not and can not have a least
element. What I not yet understand is that an empty set can house more
than zero elements, in fact more than one.

But with this premise accepted, set theory is certainly not provably
inconsistent.

Regards, WM

[William Hughes]

On Mar 16, 7:10 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 16 Mrz., 18:10, William Hughes <wpihug...@gmail.com> wrote:
>
> > > Ok, I understand. Anyhow, if the number of lines is not empty, then
> > > there must remain at least one line as a necessary line.
>
> > Not a particular line.  This is similar to
> > the case where any set of lines with an unfindable
> > last line has at least one "necessary" findable line.
> > This line has a line number in the original
> > list but we can choose the  "necessary"
> > findable line to have any line number we want.
>
> No, it is always the last line. We call it unfindable or unfixable
> because as soon as we have found it, it is no longer the last line.

Note, that I am not talking about the unfindable line,
but the "necessary" findable line. We can choose this

[WM]

In potential infinity there is no necessary line except the last one.
We know that with certainty from induction. Every found and fixed line
n cannot be necessary, because the next line contains it.

Everything that is in the list

1
1, 2
1, 2, 3
...

1, 2, 3, ..., n
is in the last line. Alas as soon as you try to fix it, it is no
longer the last line.

Think of the time. What is "now"? As soon as you try to fix it, it is
past. In time you can predict the development of clocks. In lists
there is no such smooth, predictable evolution. Will the next line
added to above list be n+1, or n^2 or n^n^n^n (all those of course
also including n+1 and its followers? There are no limits. But as soon
as we look onto the last line, we get the idea of another one and that
will add one or many lines to the list.

Regards, WM

[Virgil]

In article
<6658f152-b433-4d3e...@g16g2000vbf.googlegroups.com>,

All the lines one finds that way are findable,
at least outside Wolkenmuekenheim.


***********************************************************************

WM has frequently claimed that a mapping from the set of all infinite
binary sequences to the set of paths of a CIBT is a linear mapping.
In order to show that such a mapping is a linear mapping, WM must first
show that the set of all binary sequences is a vector space and that the
set of paths of a CIBT is also a vector space, which he has not done and
apparently cannot do, and then show that his mapping satisfies the
linearity requirement that
f(ax + by) = af(x) + bf(y),
where a and b are arbitrary members of a field of scalars and x and y
are f(x) and f(y) are vectors in suitable linear spaces.

By the way, WM, what are a, b, ax, by and ax+by when x and y are binary
sequences?

If a = 1/3 and x is binary sequence, what is ax ?
and if f(x) is a path in a CIBT, what is af(x)?

Until these and a few other issues are settled, WM will still have
failed to justify his claim of a LINEAR mapping from the set (but not
yet proved to be vector space) of binary sequences to the set (but not
yet proved to be vector space) of paths ln a CIBT.

Just another of WM's many wild claims of what goes on in his WMytheology
that he cannot back up.
--

[Ralf Bader]

WM wrote:

> On 16 Mrz., 19:26, William Hughes <wpihug...@gmail.com> wrote:

>> On Mar 16, 7:10ï¿œpm, WM <mueck...@rz.fh-augsburg.de> wrote:
>>
>> > On 16 Mrz., 18:10, William Hughes <wpihug...@gmail.com> wrote:
>>
>> > > > Ok, I understand. Anyhow, if the number of lines is not empty, then
>> > > > there must remain at least one line as a necessary line.
>>

>> > > Not a particular line. ï¿œThis is similar to

>> > > the case where any set of lines with an unfindable
>> > > last line has at least one "necessary" findable line.
>> > > This line has a line number in the original

>> > > list but we can choose the ￜ"necessary"

>> > > findable line to have any line number we want.
>>
>> > No, it is always the last line. We call it unfindable or unfixable
>> > because as soon as we have found it, it is no longer the last line.
>>
>> Note, that I am not talking about the unfindable line,

>> but the "necessary" findable line. ï¿œWe can choose this

>> line to have any line number we want
>
> In potential infinity there is no necessary line except the last one.
> We know that with certainty from induction. Every found and fixed line
> n cannot be necessary, because the next line contains it.
>
> Everything that is in the list
> 1
> 1, 2
> 1, 2, 3
> ...
> 1, 2, 3, ..., n
> is in the last line. Alas as soon as you try to fix it, it is no
> longer the last line.
>
> Think of the time. What is "now"? As soon as you try to fix it, it is
> past. In time you can predict the development of clocks. In lists
> there is no such smooth, predictable evolution. Will the next line
> added to above list be n+1, or n^2 or n^n^n^n (all those of course
> also including n+1 and its followers? There are no limits. But as soon
> as we look onto the last line, we get the idea of another one and that
> will add one or many lines to the list.
>
> Regards, WM

Will this braindead nonsense appear in the next "research report" of that
crazy so-called "university" where you work as a stupefactor of the
pitiable students?

[Virgil]

In article
<a92fd087-42fa-473d...@a14g2000vbm.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 16 Mrz., 00:39, Virgil <vir...@ligriv.com> wrote:
>
> > > Let's first prove that already two cannot be necessary by the fact
> > > that two always can be replaced by one of them without changing the
> > > contents. Then it is clear that two or more cannot be necessary and
> > > from this immediately follows that they also cannot be sufficient.
> >
> > Enough more than two line can be necessary and can be sufficient,
>
> Do you agree that every non-empty set of line-numbers contains a least
> element?

At least outside Wolkenmuekenheim that is the case.

>
>
> > > Wrong. Why do you resist to apply logic?
> >
> > A positive finite number of lines in necessary
>
> A positive finite number of lines contains a least element.

True, but irrelevant to the issue of a set of lines covering d.

>
> > We have the choice between 1 line (in potential
> >
> > > infinity) and 0 lines (in actual infinity).
> >
> > Is that a Royal "We"?
>

> No it includes evlerybody, many don't know though.

So that WM claims that in WMytheology, 1 line covers d but everywhere
else 0 lines cover d?

Does that line that WM claims covers d in WMytheology have a successor
line in WMytheology, WM?

It has a successor line everywhere else.

[Virgil]

In article
<ddb60259-5585-407a...@k14g2000vbv.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 15 Mrz., 23:27, William Hughes <wpihug...@gmail.com> wrote:
> > On Mar 15, 8:34�pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > Let's first prove that already two cannot be necessary by the fact
> > > that two always can be replaced by one of them without changing the
> > > contents.
> >
> > This is true but the fact that the two lines are
> > necessary has nothing to do with their contents. �Two lines
> > cannot be replaced by one of them without changing the number
> > of lines.
>
> Why should line-numbers be changed? Perhaps we are misunderstanding
> each other.

It appears that WM does misaunderstand, since WH did not mention , or
even imply that any line NUMBERS be changed.

>
> > Consider the case is potential infinity.

Whyever should we consider what is counter to our reality?
You are free to wallow in your own version of reality, but have not the
power to impose it on ayone else.

>
> But I would ask you in advance: Do you agree that every non-empty set
> of natural numbers (including line-numbers) has a smallest element? Or
> do you believe that here the exception proves the rule?

That is standard, but what is not is your claim that an inductive set
must contain a largest, though unfindable/inaccessible, member.
e
How are your unfindable naturals any different than inaccessible reals?

[Virgil]

In article
<5a32716b-160e-41b4...@y9g2000vbb.googlegroups.com>,

On the other hand, such a set of lines cannot have a largest element, at
least outside of Wolkenmuekenheim, since, at least outside of
Wolkenmuekenheim, for every line is a proper subset of infinitely many
successor lines.

>
> Do you think that this is unmathematical?

What goes on in Wolkenmuekenheim certainly is, alt least by the
standards of real mathematics.

[WM]

On 16 Mrz., 20:46, Virgil <vir...@ligriv.com> wrote:

> > > A positive finite number of lines in necessary
>
> > A positive finite number of lines contains a least element.
>
> True, but irrelevant to the issue of a set of lines covering d.

The lines ofthe set have numbers. These numbers can be unioned into a
set of natural numbers. This set has a least element! But we cannot
find it, because Virgil calls it "irrelevant".

>
>
>
> > > We have the choice between 1 line (in potential
>
> > > > infinity) and 0 lines (in actual infinity).
>
> > > Is that a Royal "We"?
>
> > No it includes evlerybody, many don't know though.
>
> So that WM claims that in WMytheology, 1 line covers d but everywhere
> else 0 lines cover d?

No, Ithis proves that in the land of actual infinity, that you call
"everywhere", something goes wrong. I shouldn't be surprised to see
that the reason is ctual infinity.

>
> ***********************************************************************
>
> WM has frequently claimed that a mapping from the set of all infinite
> binary sequences to the set of paths of a CIBT is a linear mapping.

Proven in Matheology § 226.

Regards, WM

[Virgil]

In article
<ad4f8623-0e9a-4b07...@hl5g2000vbb.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 16 Mrz., 18:10, William Hughes <wpihug...@gmail.com> wrote:
>
> > > Ok, I understand. Anyhow, if the number of lines is not empty, then
> > > there must remain at least one line as a necessary line.
> >
> > Not a particular line. �This is similar to
> > the case where any set of lines with an unfindable
> > last line has at least one "necessary" findable line.
> > This line has a line number in the original
> > list but we can choose the �"necessary"
> > findable line to have any line number we want.
>
> No, it is always the last line.

WH is saying that in any set of lines containing an unfindable line
there must also be a findable line.

> We call it unfindable or unfixable
> because as soon as we have found it, it is no longer the last line.

If finding it makes it not what it is supposed to be, the how does one
prove that any such thing exists?

It seems that as soon as you even try to refer to it, it is no longer
what you want it to be.

***********************************************************************

WM has frequently claimed that a mapping from the set of all infinite
binary sequences to the set of paths of a CIBT is a linear mapping.

[Virgil]

In article
<f23c81fd-2463-4d3f...@z4g2000vbz.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 16 Mrz., 19:26, William Hughes <wpihug...@gmail.com> wrote:
> > On Mar 16, 7:10�pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > On 16 Mrz., 18:10, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > > Ok, I understand. Anyhow, if the number of lines is not empty, then
> > > > > there must remain at least one line as a necessary line.
> >
> > > > Not a particular line. �This is similar to
> > > > the case where any set of lines with an unfindable
> > > > last line has at least one "necessary" findable line.
> > > > This line has a line number in the original
> > > > list but we can choose the �"necessary"
> > > > findable line to have any line number we want.
> >
> > > No, it is always the last line. We call it unfindable or unfixable
> > > because as soon as we have found it, it is no longer the last line.
> >
> > Note, that I am not talking about the unfindable line,
> > but the "necessary" findable line. �We can choose this
> > line to have any line number we want
>
> In potential infinity there is no necessary line except the last one.
> We know that with certainty from induction. Every found and fixed line
> n cannot be necessary, because the next line contains it.

AS soon as something is identifies as a natural or a FIS of the set of
naturals, it has a successor. It cannot be either a natural nor a FIS of
the naturals without a successor. at least by any standard definition of
naturals.

Can WM provide an definition for natural numberss which doe not state,
or at least imply, that every natural must have a successor natural?

>
> Everything that is in the list
> 1
> 1, 2
> 1, 2, 3
> ...
> 1, 2, 3, ..., n
> is in the last line. Alas as soon as you try to fix it, it is no
> longer the last line.

Thus it is unfixable that where there is a last line there are not all
lines nor all naturals.

>
> Think of the time. What is "now"? As soon as you try to fix it, it is
> past. In time you can predict the development of clocks. In lists
> there is no such smooth, predictable evolution. Will the next line
> added to above list be n+1, or n^2 or n^n^n^n (all those of course
> also including n+1 and its followers? There are no limits. But as soon
> as we look onto the last line, we get the idea of another one and that
> will add one or many lines to the list.

So that the process is endless.

Mathematics outside of Wolkenmuekenheim deals successfully with endless
processes all the time, but inside Wolkenmuekenheim, they are apparently
totally verbotten.

[WM]

On 16 Mrz., 21:07, Virgil <vir...@ligriv.com> wrote:

> > We call it unfindable or unfixable
> > because as soon as we have found it, it is no longer the last line.
>
> If finding it makes it not what it is supposed to be, the how does one
> prove that any such thing exists?

Simply by observing that otherwise, there must be a set with at least
two natural numbers, both of which do not belong to the set.

>
> It seems that as soon as you even try to refer to it, it is no longer
> what you want it to be.

Correct. That feature have potential infinity and ending of the past
in common

Regards, WM

[WM]

On 16 Mrz., 21:19, Virgil <vir...@ligriv.com> wrote:

> > In potential infinity there is no necessary line except the last one.
> > We know that with certainty from induction. Every found and fixed line
> > n cannot be necessary, because the next line contains it.
>
> AS soon as something is identifies as a natural or a FIS of the set of
> naturals, it has a successor. It cannot be either a natural nor a FIS of
> the naturals without a successor. at least by any standard definition of
> naturals.

As soon as a second becomes presence, it has a successor. It cannot be
presence. Nevertheless presence exists.

>
> Can WM provide an definition for natural numberss which doe not state,
> or at least imply, that every natural must have a successor natural?

Numbers are creations of the mind. Without minds there are no numbers.

>
> > Everything that is in the list
> > 1
> > 1, 2
> > 1, 2, 3
> > ...
> > 1, 2, 3, ..., n
> > is in the last line. Alas as soon as you try to fix it, it is no
> > longer the last line.
>
> Thus it is unfixable that where there is a last line there are not all
> lines nor all naturals.
>
>
>
> > Think of the time. What is "now"? As soon as you try to fix it, it is
> > past. In time you can predict the development of clocks. In lists
> > there is no such smooth, predictable evolution. Will the next line
> > added to above list be n+1, or n^2 or n^n^n^n (all those of course
> > also including n+1 and its followers? There are no limits. But as soon
> > as we look onto the last line, we get the idea of another one and that
> > will add one or many lines to the list.
>
> So that the process is endless.
>
> Mathematics outside of Wolkenmuekenheim  deals successfully with endless
> processes all the time,

but you are not able to write aleph_0 digits of a real numbers like
1/9. You can only use finite definitions to determine the limit. That
is the successful dealing of mathematics with infinity. The belief,
however, that "there are" aleph_0 digits, does not belong to
mathematics.

Regards, WM

[William Hughes]

On Mar 16, 7:38 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 16 Mrz., 19:26, William Hughes <wpihug...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On Mar 16, 7:10 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > On 16 Mrz., 18:10, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > > Ok, I understand. Anyhow, if the number of lines is not empty, then
> > > > > there must remain at least one line as a necessary line.
>
> > > > Not a particular line.  This is similar to
> > > > the case where any set of lines with an unfindable
> > > > last line has at least one "necessary" findable line.
> > > > This line has a line number in the original
> > > > list but we can choose the  "necessary"
> > > > findable line to have any line number we want.
>
> > > No, it is always the last line. We call it unfindable or unfixable
> > > because as soon as we have found it, it is no longer the last line.
>
> > Note, that I am not talking about the unfindable line,
> > but the "necessary" findable line.  We can choose this
> > line to have any line number we want
>
> In potential infinity there is no necessary line except the last one.
> We know that with certainty from induction. Every found and fixed line
> n cannot be necessary, because the next line contains it.

And yet you agree that for a set
of lines to contain an unfindable line it is necessary
that it contain at least two findable lines.

[Virgil]

In article
<54c7bd7a-5812-42a2...@g8g2000vbf.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 16 Mrz., 20:46, Virgil <vir...@ligriv.com> wrote:
>
> > > > A positive finite number of lines in necessary
> >
> > > A positive finite number of lines contains a least element.
> >
> > True, but irrelevant to the issue of a set of lines covering d.
>
> The lines ofthe set have numbers. These numbers can be unioned into a
> set of natural numbers. This set has a least element! But we cannot
> find it, because Virgil calls it "irrelevant".

WM may not be able to find it but anyone with a brain can, it is the
first natural number, 0 or 1 depending on which model one uses, which is
in EVERY non-empty union of lines.

> >
> >
> >
> > > > We have the choice between 1 line (in potential
> >
> > > > > infinity) and 0 lines (in actual infinity).
> >
> > > > Is that a Royal "We"?
> >
> > > No it includes evlerybody, many don't know though.

But as it only holds in Wolkenmuekenheim and nowhere els, it does not
include everybody, but only those, like WM, entombed in
Wolkenmuekenheim.

> >
> > So that WM claims that in WMytheology, 1 line covers d but everywhere
> > else 0 lines cover d?
>
> No

Finally WM got something right!

> >
> > ***********************************************************************
> >
> > WM has frequently claimed that a mapping from the set of all infinite
> > binary sequences to the set of paths of a CIBT is a linear mapping.
>
> Proven in Matheology � 226.

Not to the satisfaction of anyone outside Wolkenmuekenheim.

WM has not shown that either of the sets has the form of a linear space
nor that the mapping between them is, besides a bijection, also a linear
mapping.

There is a way to do it but not at all the way WM has claimed.

And not a way that WM is mathematician enough to find on his own.
--

[Virgil]

In article
<3711021c-d3eb-4fa3...@a8g2000vbx.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 16 Mrz., 21:19, Virgil <vir...@ligriv.com> wrote:
>
> > > In potential infinity there is no necessary line except the last one.
> > > We know that with certainty from induction. Every found and fixed line
> > > n cannot be necessary, because the next line contains it.
> >
> > AS soon as something is identifies as a natural or a FIS of the set of
> > naturals, it has a successor. It cannot be either a natural nor a FIS of
> > the naturals without a successor. at least by any standard definition of
> > naturals.
>
> As soon as a second becomes presence, it has a successor. It cannot be
> presence. Nevertheless presence exists.

Thus in WMytheology one must have the existence of non-existing objects.

I prefer infinities to WM's need for having what one does not have.

> >
> > Can WM provide an definition for natural numberss which doe not state,
> > or at least imply, that every natural must have a successor natural?
>
> Numbers are creations of the mind. Without minds there are no numbers.

Which is not a relevant answer.

Can WM provide an definition for natural numberss which doe not state,
or at least imply, that every natural must have a successor natural?
> >

> > > Everything that is in the list
> > > 1
> > > 1, 2
> > > 1, 2, 3
> > > ...
> > > 1, 2, 3, ..., n
> > > is in the last line. Alas as soon as you try to fix it, it is no
> > > longer the last line.
> >
> > Thus it is unfixable that where there is a last line there are not all
> > lines nor all naturals.
> >

> > Mathematics outside of Wolkenmuekenheim �deals successfully with endless
> > processes all the time,
>
> but you are not able to write aleph_0 digits of a real numbers like
> 1/9.

So what? There are lot of things in mathematics one cannot do, but that
should not keep us from doing what we can do, the way you would limit us.



######################################################################



WM has frequently claimed that his mapping from the set of all infinite

binary sequences to the set of paths of a CIBT is a linear mapping.

In order to show that such a mapping is a linear mapping, WM must first
show that the set of all binary sequences is a vector space and that the
set of paths of a CIBT is also a vector space, which he has not done and
apparently cannot do, and then show that his mapping satisfies the
linearity requirement that
f(ax + by) = af(x) + bf(y),

where a and b are arbitrary members of the field of scalars and x and y
and f(x) and f(y) are arbitrary members of suitable linear spaces.


While this is possible, and fairly trivial for a competent mathematician
to do, WM has not yet been able to do it.

But frequently claims to have already done it.
--

[Virgil]

In article
<923672fe-e549-4fb8...@z4g2000vbz.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 16 Mrz., 21:07, Virgil <vir...@ligriv.com> wrote:
>
> > > We call it unfindable or unfixable
> > > because as soon as we have found it, it is no longer the last line.
> >
> > If finding it makes it not what it is supposed to be, the how does one
> > prove that any such thing exists?
>
> Simply by observing that otherwise, there must be a set with at least
> two natural numbers, both of which do not belong to the set.

Non Sequitur, at least outside WMytheology.

Where actually infinite set of naturls is allowed, nothing like what WQM
demands is needed or even possible.

> >
> > It seems that as soon as you even try to refer to it, it is no longer
> > what you want it to be.
>
> Correct. That feature have potential infinity and ending of the past

> in commonSo potentaial infinity is never what you want it to be?

It is certainly never what we want, so lets dump it as a bad idea.





######################################################################



WM has frequently claimed that his mapping from the set of all infinite

binary sequences to the set of paths of a CIBT is a linear mapping.

In order to show that such a mapping is a linear mapping, WM must first
show that the set of all binary sequences is a vector space and that the
set of paths of a CIBT is also a vector space, which he has not done and
apparently cannot do, and then show that his mapping satisfies the
linearity requirement that
f(ax + by) = af(x) + bf(y),

[fom]

On 3/16/2013 3:19 PM, Virgil wrote:
> In article
> <f23c81fd-2463-4d3f...@z4g2000vbz.googlegroups.com>,
> WM <muec...@rz.fh-augsburg.de> wrote:

<snip>

>>
>> Think of the time. What is "now"? As soon as you try to fix it, it is
>> past. In time you can predict the development of clocks. In lists
>> there is no such smooth, predictable evolution. Will the next line
>> added to above list be n+1, or n^2 or n^n^n^n (all those of course
>> also including n+1 and its followers? There are no limits. But as soon
>> as we look onto the last line, we get the idea of another one and that
>> will add one or many lines to the list.
>
> So that the process is endless.
>
> Mathematics outside of Wolkenmuekenheim deals successfully with endless
> processes all the time, but inside Wolkenmuekenheim, they are apparently
> totally verbotten.

And, even when the endless mathematics is restricted to
potential *feasibility* as with Markov where the quantifiers
are properly reinterpreted to explicitly account for each
materially given constructive object.

[fom]

On 3/16/2013 4:37 PM, WM wrote:
> On 16 Mrz., 21:19, Virgil <vir...@ligriv.com> wrote:
>
>>> In potential infinity there is no necessary line except the last one.
>>> We know that with certainty from induction. Every found and fixed line
>>> n cannot be necessary, because the next line contains it.
>>
>> AS soon as something is identifies as a natural or a FIS of the set of
>> naturals, it has a successor. It cannot be either a natural nor a FIS of
>> the naturals without a successor. at least by any standard definition of
>> naturals.
>
> As soon as a second becomes presence, it has a successor.

And what fantasy is this?

The successor to the present has existential form but
has not yet happened.

That is not the Kantian aprioriticity of time.

That is not the Hegelian becoming of the present.

It is the unfounded object of unjustifiable belief.

It is the pot calling the kettle black.

[WM]

Please do not intermingle the facts.
If we go through the list of FISONs
1
1,2
1,2,3
...
then we can drop every FISON. We maintain all naturals "that are in
the list" if only the last line is maintained. This is like time. If
you only preserve the last second, then you have access to everything
(including memories) that existed in this last second.

Contrary to this natural although unfamiliar opinon, there are
defenders of actual infinity.
They have to claim that all natural numbers that ever can exist,
already exist in the list. This implies closure, i.e., it is
impossible to have a natural outside of the list. The list is
complete. But they deny that the list has a last line. And there comes
the inconsistency: Completeness requires and end-signal, at least in a
scientific theory that should be falsifyable or verifyable.

However, in fact nobody claims a last line. The only alternative, in
actual infinity, is that all naturals are there, but not in one line
but in two or more lines (unless you want to claim that they are in
any empty line). And this claim is contradicted by the construction
principle of the list.

What is difficult to understand?

Regards, WM

[WM]

On 17 Mrz., 00:25, Virgil <vir...@ligriv.com> wrote:

> > > Can WM provide an definition for natural numberss which doe not state,
> > > or at least imply, that every natural must have a successor natural?
>
> > Numbers are creations of the mind. Without minds there are no numbers.
>
> Which is not a relevant answer.

By definition of a matheologian.
>
> Can WM provide an definition for natural numbers which doe not state,

> or at least imply, that every natural must have a successor natural?

It is always stated or at least implicitly assumed in classical
mathematics that we are able to add 1. In reality this is an erroneous
assumption as has been shown in MatheRealism.

>
> > you are not able to write aleph_0 digits of a real numbers like 1/9.
>
> So what? There are lot of things in mathematics one cannot do, but that
> should not keep us from doing what we can do, the way you would limit us.

But the claim of matheology is that all digits exist and can be
determined - for instance all digits of pi. And that is simply wrong,
because you never get more than finitely many, such that always
infinitely many must remain unknown.

Of course distinguishing the elements of uncountable sets requires
that possibility. And if it turns out impossible, then prayers are
issued - like the axiom of choice.
Such praying and believing is part of theology and matheology.

Regards, WM

[WM]

On 17 Mrz., 00:30, Virgil <vir...@ligriv.com> wrote:

> > > > We call it unfindable or unfixable
> > > > because as soon as we have found it, it is no longer the last line.
>
> > > If finding it makes it not what it is supposed to be, the how does one
> > > prove that any such thing exists?
>
> > Simply by observing that otherwise, there must be a set with at least
> > two natural numbers, both of which do not belong to the set.
>
> Non Sequitur, at least outside WMytheology.
>
> Where actually infinite set of naturls is allowed, nothing like what WQM
> demands is needed or even possible.

In mathematics the assertion of existence of a non-empty set of
natural line-numbers implies that there is a least line-number.

Regards, WM

[WM]

It is the well known and established natural way how time passes and
how the system of human actions in time goes off.

Regards, WM

[fom]

[WM]

> It is the unfounded object of unjustifiable belief.-

Then you should love it like "the Cartesian product of non-empty sets
is non-empty".

Regards, WM

[William Hughes]

If the question is "Can a findable line be necessary"
the question of whether a finable line is needed is
certainly relevant.

[fom]

I do!

[WM]

On 17 Mrz., 12:07, William Hughes <wpihug...@gmail.com> wrote:

>
> If the question is "Can a findable line be necessary"

> the question of whether a findable line is needed is
> certainly relevant.-

Stop!

I do not force anybody to accept my model. I cannot and I do not wish
to.

The question in current mathematics was and is only this: Can a set of
more than one lines in the given list contain more than a single line?
In other words: Can more than one line be necessary?

Regards, WM

[AMeiwes]

"WM" <muec...@rz.fh-augsburg.de> wrote in message
news:f3530023-6c75-428c...@a14g2000vbm.googlegroups.com...

so your new Pope commands you to Pray to fix your Wmatheology, after you
realized it is the only way......

[Virgil]

In article
<4f0d5742-ef15-4427...@h14g2000vbe.googlegroups.com>,

Which, while true, is, as usual and as expected, irrelevant!

While there may be natural numbers which are, in some sense or other,
unfindable, there is nothing which is simultaneously a natural number
and has no successor natural number.

At least not outside WOLKENMUEKENHEIM.




######################################################################



WM has frequently claimed that HIS mapping from the set of all infinite

binary sequences to the set of paths of a CIBT is a linear mapping.

In order to show that such a mapping is a linear mapping, WM would first
have to show that the set of all binary sequences is a linear space
(which he has not done and apparently cannot do) and that the set of
paths of a CIBT is also a vector space (which he also has not done and
apparently cannot do) and then show that his mapping, say f, satisfies

the linearity requirement that f(ax + by) = af(x) + bf(y),
where a and b are arbitrary members of the field of scalars and x and y
and f(x) and f(y) are arbitrary members of suitable linear spaces.


While this is possible, and fairly trivial for a competent mathematician
to do, WM has not yet been able to do it.

But frequently claims already to have done it.
--

[Virgil]

In article
<f3530023-6c75-428c...@a14g2000vbm.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 17 Mrz., 00:25, Virgil <vir...@ligriv.com> wrote:
>

> > > > Can WM provide any definition for natural numberss which doe not state,

> > > > or at least imply, that every natural must have a successor natural?
> >
> > > Numbers are creations of the mind. Without minds there are no numbers.
> >
> > Which is not a relevant answer.
>
> By definition of a matheologian.

Which is a non-response to my original question:
Can WM provide any definition for natural numberss which doe not state,

or at least imply, that every natural must have a successor natural?

WM's failure to respond positively I take as a "no" answer.

> >
> >
> > Can WM provide an definition for natural numbers which doe not state,
> > or at least imply, that every natural must have a successor natural?
>
> It is always stated or at least implicitly assumed in classical
> mathematics that we are able to add 1. In reality this is an erroneous
> assumption as has been shown in MatheRealism.

Unless you can produce such a natural, which even in your alleged
"mathrealism" you have not done, your claim is, as always, unfounded.

> >
> > > you are not able to write aleph_0 digits of a real numbers like 1/9.
> >
> > So what? There are lot of things in mathematics one cannot do, but that
> > should not keep us from doing what we can do, the way you would limit us.
>
> But the claim of matheology is that all digits exist and can be
> determined - for instance all digits of pi.

I do not make the claim that all the digits of the number pi can be
found, but I do claim that the number pi has a definition, as do
infinitely many other reals whose exact decimal representations cannot
be found.

E. g., the square roots of each of the infinitely many primes are all
defined, though none can be given exactly as decimals.






######################################################################



WM has frequently claimed that HIS mapping from the set of all infinite

binary sequences to the set of paths of a CIBT is a linear mapping.

In order to show that such a mapping is a linear mapping, WM would first
have to show that the set of all binary sequences is a linear space
(which he has not done and apparently cannot do) and that the set of
paths of a CIBT is also a vector space (which he also has not done and
apparently cannot do) and then show that his mapping, say f, satisfies

the linearity requirement that f(ax + by) = af(x) + bf(y),
where a and b are arbitrary members of the field of scalars and x and y
and f(x) and f(y) are arbitrary members of suitable linear spaces.


While this is possible, and fairly trivial for a competent mathematician
to do, WM has not yet been able to do it.

[Virgil]

In article
<39d6bf7c-843b-4999...@he10g2000vbb.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 16 Mrz., 23:12, William Hughes <wpihug...@gmail.com> wrote:

> > And yet you agree that for a set
> > of lines to contain an unfindable line it is necessary
> > that it contain at least two findable lines.
>
> Please do not intermingle the facts.

Why should WH bhe prohibited from doing what WM does so regularly?

> If we go through the list of FISONs
> 1
> 1,2
> 1,2,3
> ...

But such an ellipsis represents a list that does not ever terminate.






We maintain

You maintain nonsense!

>
> However, in fact nobody claims a last line. The only alternative, in
> actual infinity, is that all naturals are there, but not in one line
> but in two or more lines (unless you want to claim that they are in
> any empty line). And this claim is contradicted by the construction
> principle of the list.

Not outside Wolkenmuekenheim.

Outside Wolkenmuekenheim one can see that in order to include all
naturals in such aa set of lines one needs more than any finite set of
lines, because for each line included, one needs some (but not
necessarily the immediate) successor line.

>
> What is difficult to understand?

That is our question, and WM doesn't have any good answer to it.
>
> Regards, WM

[Virgil]

In article
<077d0447-e3a0-4154...@gp5g2000vbb.googlegroups.com>,

If those lines are FISONs. any finite set of them is limited to the
finitely many naturals contained in the largest of them, so one needs a
non-empty set of FISONs that does not have a largest FISON, which can
only be achieved but having an actually infinite set of FISONs.

Which means that in WM's world, it can never happen at all.

[Virgil]

In article
<ded07b82-5853-4a1e...@j9g2000vbz.googlegroups.com>,

Mathematical truth is independent of time.

What was true yesterday will be true tomorrow
and what was false yesterday will almost always be false tomorrow.

Of course what had not yet been proved yesterday may be proved by
tomorrow, but it was still as true yesterday as it will be tomorrow.

So that WM's time image is an irrelevancy.

And similarly, the natural numbers of any tomorrow were already natural
numbers in every yesterday.






######################################################################



WM has frequently claimed that HIS mapping from the set of all infinite

binary sequences to the set of paths of a CIBT is a linear mapping.

In order to show that such a mapping is a linear mapping, WM would first
have to show that the set of all binary sequences is a linear space
(which he has not done and apparently cannot do) and that the set of
paths of a CIBT is also a vector space (which he also has not done and
apparently cannot do) and then show that his mapping, say f, satisfies

the linearity requirement that f(ax + by) = af(x) + bf(y),

[Virgil]

In article
<9333e03e-1364-445e...@z4g2000vbz.googlegroups.com>,

I should very much be interested in seeing WM's example of a Cartesian
product of non-empty sets that is empty, which WM should be able to
construct if the axiom of choice is as false as WM claims it to be.

But then WM cannot even prove that a mapping which he claims is linear
is actually linear.




######################################################################



WM has frequently claimed that HIS mapping from the set of all infinite

binary sequences to the set of paths of a CIBT is a linear mapping.

In order to show that such a mapping is a linear mapping, WM would first
have to show that the set of all binary sequences is a linear space
(which he has not done and apparently cannot do) and that the set of
paths of a CIBT is also a vector space (which he also has not done and
apparently cannot do) and then show that his mapping, say f, satisfies

the linearity requirement that f(ax + by) = af(x) + bf(y),

[WM]

On 17 Mrz., 21:48, Virgil <vir...@ligriv.com> wrote:

> Mathematical truth is independent of time.

In fact??? Amazing! After Cantor's list has been diagonalized, it is
possible to include all diagonals into the list. But someone has
forbidden to change the list after time t_0 when the diagonalizers
start to do their work.

Regards, WM

[William Hughes]

You are contradicting yourself.

You say there are no necessary findable lines
because of the last line (an unfindable line)

You say that if a set of lines contains an unfindable
line it is necessary that there are
two findable lines.

[WM]

On 17 Mrz., 22:39, William Hughes <wpihug...@gmail.com> wrote:
> You are contradicting yourself.
>
> You say there are no necessary findable lines
> because of the last line (an unfindable line)

In pot. inf. there is always a last line. It is unfindable or
unfixable. But it is the only necessary line in potential infinity
because the list does not contain more (because there *is* not more).

But I do not wish to discuss potential infinity but actual infinity
here.

>
> You say that if a set of lines contains an unfindable
> line it is necessary that there are
> two findable lines.

No. I say that in actual infinity a list contains all natural numbers.
But they cannot be in one line, because there is no actually infinite
line. This is a contradiction.

Please kindly note: Even if my personal theory was self-contradictory,
that would not improve the situation presently adopted in mathematics.
So please concentrate on defending your position.

Regards, WM

[William Hughes]

On Mar 17, 10:49 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 17 Mrz., 22:39, William Hughes <wpihug...@gmail.com> wrote:

> > You say that if a set of lines contains an unfindable
> > line it is necessary that there are
> > two findable lines.
>
> No.

Oh, so there can be a set of lines that contains an unfindable
line but not two findable lines ?!?

[fom]

On 3/17/2013 3:48 PM, Virgil wrote:
> In article
> <ded07b82-5853-4a1e...@j9g2000vbz.googlegroups.com>,
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>> On 17 Mrz., 08:18, fom <fomJ...@nyms.net> wrote:
>>> On 3/16/2013 4:37 PM, WM wrote:
>>>
>>>> On 16 Mrz., 21:19, Virgil <vir...@ligriv.com> wrote:
>>>
>>>>>> In potential infinity there is no necessary line except the last one.
>>>>>> We know that with certainty from induction. Every found and fixed line
>>>>>> n cannot be necessary, because the next line contains it.
>>>
>>>>> AS soon as something is identifies as a natural or a FIS of the set of
>>>>> naturals, it has a successor. It cannot be either a natural nor a FIS of
>>>>> the naturals without a successor. at least by any standard definition of
>>>>> naturals.
>>>
>>>> As soon as a second becomes presence, it has a successor.
>>>
>>> And what fantasy is this?
>>>
>>> The successor to the present has existential form but
>>> has not yet happened.
>>>
>>> That is not the Kantian aprioriticity of time.
>>>
>>> That is not the Hegelian becoming of the present.
>>>
>>> It is the unfounded object of unjustifiable belief.
>>
>> It is the well known and established natural way how time passes and
>> how the system of human actions in time goes off.
>
> Mathematical truth is independent of time.

Well that depends on how a philosophy which makes
such a statement addresses the issue.

Frege specifically addresses the sense expressed
by WM:

"Next there may be those who will
prefer some other definition as
being more natural, as for example
the following:

if starting from x we transfer our
attention continually from one object
to another to which it stands in
relation phi, and if by this procedure
we can finally reach y, then we say
that y follows in the phi-series after
x.

"Now this describes a way of discovering
that y follows, it does not define what is
meant by y's following. Whether as our
attention shifts, we reach y may depend
on all sorts of subjective contributory
factors, for example, on the amount of
time at our disposal or on the extent of
our familiarity with the things concerned.
Whether y follows in the phi-series of
x has nothing to do with our attention
and the circumstances in which we
transfer it; on the contrary, it is a
question of fact, just as much as it is
a fact that a green leaf reflects light
rays of certain wavelengths, whether or
not these fall into my eye and give rise
to a sensation, and a fact that a
grain of salt is soluble in water whether
or not I drop it into water and observe
the result, and a further fact that
it remains still soluble even when it
is utterly impossible for me to make
any experiment with it.

"My definition lifts the matter into
a new plane; it is no longer a question
of what is subjectively possible but
of what is objectively definite. For
in literal fact, that one proposition
follows from certain others is something
objective, something independent of the
laws that govern the movements of our
attention, something to which it is
immaterial whether we actually draw the
conclusion or not. What I have provided
is a criterion which decides in every case
the question "Does it follow after?"
wherever it can be put; and however much
in particular cases we may prevented by
extraneous difficulties from actually
reaching a decision, that is irrelevant
to the fact itself.



Although Frege eventually retracted
his own definition, what he is saying
here is that defined relations relative
to a defined logic constitute the matter
of an objective mathematics.


To understand the distinction, one may
contrast Frege with Weyl. The latter
is, at least, tentatively willing to
admit a set theoretic ground that does
not yield the transfinite. He says this
in reference to Dedekind:

"A set-theoretic treatment of the natural
numbers such as that offered by Dedekind
may indeed contribute to the systematization
of mathematics; but it must not be
allowed to obscure the fact that our
grasp of the basic concepts of set theory
depends on a prior intuition of iteration
and of the sequence of natural numbers."

Let me give credit to WM for rejecting
Dedekind. He has shown enough consistency
to realize that a Dedekindian ground is
a ground that fixes the successor relation
with respect to a system that constitutes
a completed infinity. Weyl apparently
misses the inconsistency of his position
out of desire to reject transfinite
arithmetic.

But, a few pages earlier, Weyl makes an
interesting statement concerning the nature
of "objective" fact.

Note the explicit rejection of logic
and definition in his statement,

"Therefore, how two sets (in contrast to
properties) are defined (on the basis of
the primitive properties and relations
and individual objects exhibited by means
of the principles of section 2) does not
determine their identity. Rather, an
objective fact which is not decipherable
from the definition in a purely logical
way is decisive; namely, whether each
element of the one set is an element
of the other, and conversely. [...]"


So, as a reader of this statement, I
am first expected to reject prior
definitions and to reject logical
relations. Then, I am expected to
understand the discursive assertion
explaining what it is that cannot
be explained.

However, I am to understand that this
is sensible with respect to some
other prior principles explained
elsewhere. And, I am to understand
that what cannot be explained to
me can sensibly be expressed as
a rule.

The statement goes on to say,

"Moreover, we see that the description
of a finite set in individual terms
is, considered formally, just a special
case of that based on a rule. For
example, if a,b,c are three objects
of our category, then

P(x)=J(xa)+J(xb)+J(xc)

is the judgement scheme of the derived
property 'being a or b or c'; and the
set having just those three objects
as its elements correspond to this
property."


What is relevant from section 2 that
I am expected to not ignore while
being told to ignore is the following:

"By simple or primitive judgment scheme
we mean those which correspond to the
individual immediately given properties
and relations. To these we add the
identity scheme J(xy) (meaning 'x is
identical to y' i.e., 'x=y')"


So, once again, the situation resolves
to the concept of "immediately given
individual properties" or the objective
fact that the purport of singular
reference suffices as an establishment
of singular reference.

And, once again, searching through these
philosophies and the definitions leads
to the fact that presentations of
Leibniz law such as

http://plato.stanford.edu/entries/identity-relative/#1

misrepresents what, in fact, Leibniz
actually wrote:

"What St. Thomas affirms on this point
about angels or intelligences ('that
here every individual is a lowest
species') is true of all substances,
provided one takes the specific
difference in the way that geometers
take it with regard to their figures."

Leibniz


Returning to how time might inform
mathematics in relation to arithmetical
progressions, there is Kant:

1)
Time is not an empirical concept
that is derived from experience.
[...]

2)
Time is a necessary representation
that underlies all intuitions
[...]

3)
The possibility of apodeictic principles
concerning the relations of time, or
of axioms of time in general is
grounded upon this a priori necessity.
[...] We should only be able to
say that common experience teaches
that this is so; not that it must be
so. These principles are valid as
rules under which alone experiences
are possible; and they instruct us
in regard to experiences, not be
means of them.

4)
Time is not a discursive, or what is
called a general concept, but a form
of pure sensible intuition.

5)
The infinitude of time signifies
nothing more than that every determinate
magnitude of time is possible only
through limitations of one single
time that underlies it."



And, should anyone who wishes to reject
the mathematical aspect of Kant's
philosophy in relation to his remarks
here on the basis of nineteenth and
twentieth century "progress" one need
only consider the author to whom George
Greene directed me to learn about why
Kant had become outdated. Boolos
writes:

"[Crispin] Wright regard's Hume's
principle as a statement whose role
is to fix the character of a certain
concept. We need not read any
contemporary theories of the a priori
into the debate between Frege and Kant.
But Frege can be thought to have carried
the day against Kant only if it has
been shown that Hume's principle is
analytic, or a truth of logic. This
has not been done. [...]

"Well. Neither Frege nor Dedekind showed
arithmetic to be a part of logic. Nor
did Russell. Nor did Zermelo or von Neumann.
Nor did the author of Tractatus 6.02 or
his follower Church. They merely shed
light on it."


And, George's recommendation seemed
particularly odd when I found that
Boolos quoted Hao Wang's remark,

"The reduction, however, cuts both
ways. It is not easy to see how
Frege can avoid the seeming frivolous
argument that if his reduction is
successful, one who believes firmly
in the synthetic character of
arithmetic can conclude that Frege's
logic is thus proved to be synthetic
rather than that arithmetic is
proved to be analytic."



So, one may clearly hold the contrary
to Virgil's statement depending on
how one views time and its relationship
to mathematical thought without,
apparently, being in too much
mathematical jeopardy.

[fom]

On 3/17/2013 4:06 AM, WM wrote:
> On 17 Mrz., 00:25, Virgil <vir...@ligriv.com> wrote:
>
>>>> Can WM provide an definition for natural numberss which doe not state,
>>>> or at least imply, that every natural must have a successor natural?
>>
>>> Numbers are creations of the mind. Without minds there are no numbers.
>>
>> Which is not a relevant answer.
>
> By definition of a matheologian.
>>
>> Can WM provide an definition for natural numbers which doe not state,
>> or at least imply, that every natural must have a successor natural?
>
> It is always stated or at least implicitly assumed in classical
> mathematics that we are able to add 1. In reality this is an erroneous
> assumption as has been shown in MatheRealism.

So, WH and Virgil do the hard work to expose
the fact that WM assume's the directed
set structure of the natural numbers in
his statements.

I do the easy work of trying to place this
into a sensible construct such as the use
of a successor in Euclid's proof that there
is no greatest prime so that WM's
unfindable line has some sort of mangled
basis in classical mathematics.

But, it is all in vain.

One can know the reality of a future moment
that has not yet happened.

But, Euclid's assumption of a successor
is as erroneous as anything Cantor,
Dedekind, Brouwer, etc. has done.

[fom]

On 3/17/2013 12:01 PM, WM wrote:
> On 17 Mrz., 12:07, William Hughes <wpihug...@gmail.com> wrote:
>
>>
>> If the question is "Can a findable line be necessary"
>> the question of whether a findable line is needed is
>> certainly relevant.-
>
> Stop!
>
> I do not force anybody to accept my model.

For that to make any sense whatsoever,
you need to define for us what you
mean by "model" since you reject the
standard mathematics by which that term
is used.

[fom]

The possibility of error in Virgil's statement
reflects his enthusiasm for mathematics without
the full breadth of the historical background.

It may not be in error depending upon his
beliefs concerning time and mathematics.

Your retort, however, neither answers the
statement nor addresses his position.

[Virgil]

In article
<0ff7b4e3-a20a-44a5...@a14g2000vbm.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 17 Mrz., 22:39, William Hughes <wpihug...@gmail.com> wrote:
> > You are contradicting yourself.
> >
> > You say there are no necessary findable lines
> > because of the last line (an unfindable line)
>
> In pot. inf. there is always a last line.

But as every line implies the actual existence of a successor line, no
such last line can exist ong enough to be seen.

> It is unfindable or
> unfixable.

An non-existable!

> But I do not wish to discuss potential infinity but actual infinity
> here.
> >
> > You say that if a set of lines contains an unfindable
> > line it is necessary that there are
> > two findable lines.
>
> No. I say that in actual infinity a list contains all natural numbers.
> But they cannot be in one line, because there is no actually infinite
> line. This is a contradiction.

Only if one claims that a FISON need not be a FISON.

>
> Please kindly note: Even if my personal theory was self-contradictory

Which it is!

> that would not improve the situation presently adopted in mathematics.
> So please concentrate on defending your position.

We define the set of naturals so that it has a unique first member and
for each member there is a successor member larger that that predecessor.

According to that definition, all WMytheology is nonsense.

[Virgil]

In article
<ba28932b-ac48-4567...@z4g2000vbz.googlegroups.com>,

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?

After each anti-dagonal of any list is found, prefix it to that list and
then the anti-diagonal to the new list is not in the new list or the old
sub-list.

This procedure always finds new lines which are non-members of any of
the prior lists of lines including all lines of any original list and
all previously found anti-diagonals of those prior lists.

WM is just not paying attention!

[Ross A. Finlayson]

On Mar 17, 4:59 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> <ba28932b-ac48-4567-8e5c-a7e9262f8...@z4g2000vbz.googlegroups.com>,

With whatever expansions EF would have, the binary antidiagonal is at
the end. Obviously, prepending .111... to the beginning is not then
EF (the function modeled by n/d, n->d, d->oo).

Bo-ring. Virgil my good man: at this rate your work will be that
post.

So, the other day, you appended to your signature, as it were, though
it's not the four lines nor is it split from the body with --, that
there was such a linear mapping, then, how is [0,1] or the CIBT or
Cantor space of the Cantor set of the sequences 2^w, a linear space?
It's simple to define operations that would be fields except for
associativity of *, distributivity, or multiplicative inverses, so I
wonder what positive input you had in mind (and that they would
otherwise satsify the vector space axioms). A simple and trivial
continuous mapping was noted.

Regards,

Ross Finlayson

[fom]

And, necessary to the proof of an algebraically closed field.

But when one 'knows' such things 'by reality' there is
no need of definition or proof.

[fom]

On 3/17/2013 7:11 PM, Ross A. Finlayson wrote:
>
> A simple and trivial
> continuous mapping was noted.
>
> Regards,
>
> Ross Finlayson
>

That is not enough Ross.

By definition, a linear map must satisfy

f(x+y) = f(x) + f(y)
f(ax) = a*f(x)

So, the domain must at least have the
structure of a module since it needs
to have an abelian addition of domain
elements and a map from the domain
into itself with a scalar multiplication.

Furthermore, it is unlikely that one
could take the scalar multiplication
to be the Galois field over two
elements since multiplication by
zero would be the zero vector and
multiplication by one would be
the identity map.

A morphism with that scalar field
could not reasonably be expected
to have a linear map with a
system of real numbers.

In order to build a scalar that
could even possibly serve this
purpose, given WM's claims related
to various finite processes, one
would have to invoke compactness
arguments involving completed
infinities.

For example, for any non-zero
sequence of zeroes and ones
that becomes eventually constant
with a trailing sequence of zeroes,

1001101000......

we can replace that sequence with
a trailing sequence of ones,

1001101111......

We want to use these forms because
of the products

1*1=1
1*0=0
0*1=0
0*0=0

Then, coordinatewise multiplications
along the trailing sequence of ones
retains a trailing sequence of ones.

In addition, on the interval

0<x<=1

we can associate 1 with the constant
sequence,

111...

Given these facts, we can now say that
a collection of infinite sequences is
"compactly admissible" if for every
finite collection of those sequences
coordinatewise multiplication yields
a sequence different from one
consisting solely of an initial
segment of zeroes followed by
an initial segment of ones.

In other words, even though

000000111...

may be representationally
equivalent to

000001000...

for some purposes, compact
admissibility has to ignore
what happens in this conversion.
The situation above is
interpreted as corresponding
with a non-compact set of
sequences.

Given this, sequences like

1000...
11000...
110000...
1101000...

yield

1111..
11111...
110111...
1101111...

whose coordinatewise product
is

1101111...

So that the original sequence
is compactly admissible.

Given a construction along these
lines, one could then think of
compactly admissible collections
as possibly forming a sequence space
as described here

http://en.wikipedia.org/wiki/Hilbert_space#Second_example:_sequence_spaces

Obviously, the compactly admissible
collections are not defined as
converging in the sense of a sequence
of partial sums.

Equally obviously, I have not done
all the work necessary to decide
whether or not this would work.

My purpose here is to explain that
the scalar multiplication would
require a construction along these
lines just to even begin to talk
about whether or not WM could
do what Virgil is asking.

[fom]

On 3/17/2013 6:59 PM, Virgil wrote:
> In article
> <ba28932b-ac48-4567...@z4g2000vbz.googlegroups.com>,
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>> On 17 Mrz., 21:48, Virgil <vir...@ligriv.com> wrote:
>>
>>> Mathematical truth is independent of time.
>>
>> In fact??? Amazing! After Cantor's list has been diagonalized, it is
>> possible to include all diagonals into the list. But someone has
>> forbidden to change the list after time t_0 when the diagonalizers
>> start to do their work.
>
> 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?

It is that problem with singular
terms again.

[Ross A. Finlayson]

> http://en.wikipedia.org/wiki/Hilbert_space#Second_example:_sequence_s...

>
> Obviously, the compactly admissible
> collections are not defined as
> converging in the sense of a sequence
> of partial sums.
>
> Equally obviously, I have not done
> all the work necessary to decide
> whether or not this would work.
>
> My purpose here is to explain that
> the scalar multiplication would
> require a construction along these
> lines just to even begin to talk
> about whether or not WM could
> do what Virgil is asking.

I looked to it that a linear mapping would need a vector space over a
field. Then basically it was found various magma(s), those being a
set equipped with an operation closed in the set, using addition being
the integer part of natural addition and multiplication the integer
part of natural multiplication. But that is not a field because it
lacks distributivity, and multiplicative inverses. Then there's the
notion to define addition-1 being the non-integer part of natural
addition, and addition-2 being the non-integer part of natural
addition, that equals one if the non-integer part is zero, so there
are two operations with that are associate, transitive, have inverses
in the field and distinct identities, but addition-2 isn't
distributive.

So, the question is, if Virgil says there exists a field over [0,1],
or the elements of the CIBT or Cantor set, there would be a continuous
function f: R_[0,1] <-> R that had a (+) b = f^-1(f(a)+f(b)) and a (*)
b = f^-1(f(a)f(b)), and that f(ab) = f(a)f(b) and f(a) + f(b) = f(a
+b).

So from an apocryphal comment that there is a linear mapping and thus
vector space and field over [0,1], I wonder how Virgil backs this
claim, as I well imagine it's not a linear function with f(0) = -oo
and f(1) = oo. (And it is.)

Then, about compact admissibility, yes there are general notions that
if N and R are compactified it's with points at infinity, then about
the form and product you mention, there is not an inverse of the
product, and I don't see it defined for all the elements of the CIBT
or Cantor set. Please feel free to further explain that.

Regards,

Ross Finlayson

[fom]

And how has one realized it?

Events that have not happened exist.

A completed future exists.

But, there is no complete infinity.

[fom]

On 3/18/2013 1:39 AM, Ross A. Finlayson wrote:
>
> So, the question is, if Virgil says there exists

First, although I have not read that signature
element too carefully, I doubt Virgil is claiming
that anything exists. WM made a claim. Virgil
is demanding proof of the claim according to the
standard meaning of the terms.

> a field over [0,1],
> or the elements of the CIBT or Cantor set, there would be a continuous
> function f: R_[0,1] <-> R that had a (+) b = f^-1(f(a)+f(b)) and a (*)
> b = f^-1(f(a)f(b)), and that f(ab) = f(a)f(b) and f(a) + f(b) = f(a
> +b).

You are making a mistake in these equations.

The multiplication in the definition,

f(x+y) = f(x) + f(y)
f(ax) = a*f(x)

is a scalar multiplication.

Of course, so much mathematics is done in
familiar number systems where the scalar
domain is related to the arithmetic of the
additive abelian group that one does not
think twice about it.

This is different. There is no definition
for multiplication of sequences from the
binary tree as sequences in the binary tree.

Linear mappings in this sense are not
immediately about continuity. Continuity
is a topological property. Linearity in
this sense is an algebraic property.

>
> So from an apocryphal comment that there is a linear mapping and thus
> vector space and field over [0,1], I wonder how Virgil backs this
> claim, as I well imagine it's not a linear function with f(0) = -oo
> and f(1) = oo. (And it is.)
>

Virgil is not making a claim. He is asking
that a claim be substantiated.

> Then, about compact admissibility

<snip>

It is a pretend construction that was
done to give you an idea of what might
be required for WM to meet Virgil's
expectations.

I would be required to do some work
to make it something you could put
on a wikipedia page....

[Virgil]

In article <mOOdnU1k7dJ5ONvM...@giganews.com>,

MY points are

(1) The bijective mapping from the set of binary sequences to the set of
paths of a Complete Infinite Binary Tree, was NOT a linear mapping as it
was originally formulated by WM.

(2) WM is not competent enough to be able to reformat it correctly .
i.e., to make it an actual and obvious linear mapping.

(3) It is not all that difficult to create an actual and obvious linear
mapping there for someone who knows something more about linear spaces
that WM does.
--

[Virgil]

In article
<ce69aaf4-ebe0-4085...@8g2000pbm.googlegroups.com>,

A linear mapping between linear spaces need not be in any sense a
continuous mapping or involve any continuity at all, as no topological
structure is required of linear spaces in general.

So Ross is, as usual, off on an irrelevant tangent again.
--

[fom]

Since you use the word "obvious" I am assuming
that I did much too much work attempting to
establish a scalar multiplication.

But, I would assume that you would be
basing it on the representations as real
numbers directly. I tried to avoid that
because of WM's finitist claims and the
abstract definition of tree paths.

[Virgil]

In article <HYednYHEPayMIdvM...@giganews.com>,

fom <fom...@nyms.net> wrote:

> Virgil is not making a claim. He is asking
> that a claim be substantiated.

Actually I did make a claim, namely that any reasonably sharp
mathematician should be able to substantiate the claim that WM made but
has so far failed to substantiate.

But it would take more imaginative creativity that I suspect WM can
summon.

If WM does not manage to find it out for himself fairly soo, which I
suspect he will not, I will publish it myself.

When I do, I suspect that WM will either claim that my way was his own
idea or try to claim that my method linear mapping is not really linear
mapping.
--

[WM]

When you remove every line as soon as you have found it, then no
findable line remains. Isn't that obvious?

However this might not be interesting for the majority of readers.
Much more interesting will be how the case of actual infinity can be
explained without contradicting the construction principle of our well-
known list.

Regards, WM

[WM]

On 18 Mrz., 00:27, fom <fomJ...@nyms.net> wrote:

> One can know the reality of a future moment
> that has not yet happened.

And one knows that that moment belongs to a finite clock-display and
to a finite maximum natural number.

Regards, WM

[WM]

On 18 Mrz., 00:30, fom <fomJ...@nyms.net> wrote:

> > I do not force anybody to accept my model.
>
> For that to make any sense whatsoever,
> you need to define for us what you
> mean by "model" since you reject the
> standard mathematics by which that term
> is used.

The word model may be used in some sciences including mathematics.
That does not force me to do anything.

Regards, WM

[WM]

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.

Both these facts are the two sides of infinity. Cantor did that false
step to choose one as more justified than the other.

Regards, WM

[WM]

On 18 Mrz., 06:28, fom <fomJ...@nyms.net> wrote:


It has been done already long ago (see Matheology § 226).
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.

Regards, WM

[WM]

On 18 Mrz., 00:59, Virgil <vir...@ligriv.com> wrote:

> In article
> <ba28932b-ac48-4567-8e5c-a7e9262f8...@z4g2000vbz.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 17 Mrz., 21:48, Virgil <vir...@ligriv.com> wrote:
>
> > > Mathematical truth is independent of time.
>
> > In fact??? Amazing! After Cantor's list has been diagonalized, it is
> > possible to include all diagonals into the list. But someone has
> > forbidden to change the list after time t_0 when the diagonalizers
> > start to do their work.
>
> 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?
>
> After each anti-dagonal of any list is found, prefix it to that list and
> then the anti-diagonal to the new list is not in the new list or the old
> sub-list.
>
> This procedure always finds new lines which are non-members of any of
> the prior lists of lines including all lines of any original list and
> all previously found anti-diagonals of those prior lists.

Every list yields another diagonal.
And every diagonal can be included in another list.

Both these facts are the two sides of infinity. Cantor did that false
step to choose one as more justified than the other.

With same right he could have argued that there are more even than odd
non-negative integers, because 0 is even and after every odd integer
there follwos an even one, just overlooking that after every even
integer there follows an odd one.

Regards, WM

[William Hughes]

On Mar 18, 12:47 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 17 Mrz., 23:05, William Hughes <wpihug...@gmail.com> wrote:
>
> > On Mar 17, 10:49 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > On 17 Mrz., 22:39, William Hughes <wpihug...@gmail.com> wrote:
> > > > You say that if a set of lines contains an unfindable
> > > > line it is necessary that there are
> > > > two findable lines.
>
> > > No.
>
> > Oh, so there can be a set of lines that contains an unfindable
> > line but not two findable lines ?!?
>
> When you remove every line as soon as you have found it, then no
> findable line remains. Isn't that obvious?

So you take a set of lines that contains an unfindable line
remove all the findable lines and end up with a set
that contains an unfindable line, but no findable line ?!?

[WM]

On 18 Mrz., 13:50, William Hughes <wpihug...@gmail.com> wrote:

> So you take a set of lines that contains an unfindable line
> remove all the findable lines and end up with a set
> that contains an unfindable line, but no findable line ?!?

If you remove every findable line, there cannot remain a findable
line, can it?

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.

Regards, WM

[fom]

What would be interesting for your readers
would see an appropriate explanation
of a constructive object.

That, being given as prior definition,
would constitute a "construction principle".

Once again, you were given an example from
Markov to see how it is done.

[fom]

So, the search for a beginning to time has been
concluded.

Not only is it less than 13.7 billion years,
it is less than the sum of "begats" discussed
at the Scope's "monkey" trial.

[fom]

Each science has its norms.

Mathematics has a well-construed notion of
model.

Per the illocutionary agreements of its practitioners,
you may use the standard characterization or provide a
new one for consideration.

How a child uses the word 'model' when
given some glue and pre-fabricated plastic
pieces is a different use of the word.

[fom]

On 3/18/2013 6:59 AM, WM 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.

What it shows "simply and exactly" is that you
do not understand the argument of Cantor diagonalization.

>
> Every list yields another diagonal.
> And every diagonal can be included in another list.

Blah, blah.

You are assuming the directed set structure of
the natural numbers. That is well-established.

>
> Both these facts are the two sides of infinity.

*ALERT*

Notice the reference to infinity in the sense
of a singular identifying term.

> Cantor did that false
> step to choose one as more justified than the other.

To be precise, Cantor to the step (truth and falsity
are irrelevant) to treat infinity with the same
justification as one treats 0.

That is why the null class trivially satisfies
the idea of a limit ordinal.

[fom]

It has not been done at all.

You may perform the requested task according to
the standard definitions used in mathematics
or you may propose new definitions to be
considered and *agreed* upon.

You have done neither.

[fom]

On 3/18/2013 7:29 AM, WM wrote:
>
> Every list yields another diagonal.
> And every diagonal can be included in another list.
>
> Both these facts are the two sides of infinity.

What these facts correspond with is the
structure of a directed set.

Directed sets are, indeed, infinite.

Absolute infinity cannot be treated as
a number.

That does not mean one cannot have a
transfinite arithmetic.

<snip>

> With same right he could have argued that there are more even than odd
> non-negative integers, because 0 is even and after every odd integer
> there follwos an even one, just overlooking that after every even
> integer there follows an odd one.

But, Cantor understood the strict reasoning inherent
to the nature of class-based logical construction.

What he could have done if he were of lesser skill
is not what he did do -- because he knew better.

[fom]

I knew this looked familiar.

So, I have some real space with an algebraic dimension
given by the finite list of its coordinates.

But, absent continuity constraints, it all fits
on one line having a single coordinate.

Damn that Dirichlet!

http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf

page 19

[WM]

Show your full ignorance of math, and by that fact justify that you
had to leave academic world, by refuting that the identity mapping of |
R on |R is an isomorphism.

Regards, WM

[Virgil]

In article
<366d66b2-1c9b-4af4...@w3g2000vba.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 18 Mrz., 00:59, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <ba28932b-ac48-4567-8e5c-a7e9262f8...@z4g2000vbz.googlegroups.com>,
> >
> > �WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 17 Mrz., 21:48, Virgil <vir...@ligriv.com> wrote:
> >
> > > > Mathematical truth is independent of time.
> >
> > > In fact??? Amazing! After Cantor's list has been diagonalized, it is
> > > possible to include all diagonals into the list. But someone has
> > > forbidden to change the list after time t_0 when the diagonalizers
> > > start to do their work.
> >
> > 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?
> >
> > After each anti-dagonal of any list is found, prefix it to that list and
> > then the anti-diagonal to the new list is not in the new list or the old
> > sub-list.
> >
> > This procedure always finds new lines which are non-members of any of
> > the prior lists of lines including all lines of any original list and
> > all previously found anti-diagonals of those prior lists.
>
> Every list yields another diagonal.

So that as soon as any list exists, a non-member of that list is proved
to exist, so the very existence of a list proves its incompleteness.

> And every diagonal can be included in another list.
>
> Both these facts are the two sides of infinity. Cantor did that false
> step to choose one as more justified than the other.

The issue is whether a list can be complete in the sense of containing
every possible sequence.

You concede that the very existence of a list demonstrates the existence
of a diagonal not listed in it.

Thus one must conclude that no such list can list everything.

>
> With same right he could have argued that there are more even than odd
> non-negative integers, because 0 is even and after every odd integer
> there follwos an even one, just overlooking that after every even
> integer there follows an odd one.

On CAN conclude that there are at least as many evens as odds from your
argument, but not that there are more of them.

To conclude that there are more of one type than another one would have
to show not only that the supposedly smaller set injects into the
supposedly larger but the supposedly larger set does NOT inject into the
supposedly smaller.

At least in the sort of proper mathematics that goes on outside of
WMytheology. What goes on inside of WMytheology is entirely up to WM.





######################################################################



WM has frequently claimed that HIS mapping from the set of all infinite
binary sequences to the set of paths of a CIBT is a linear mapping.

In order to show that such a mapping is a linear mapping, WM would first
have to show that the set of all binary sequences is a linear space
(which he has not done and apparently cannot do) and that the set of
paths of a CIBT is also a vector space (which he also has not done and
apparently cannot do) and then show that his mapping, say f, satisfies
the linearity requirement that f(ax + by) = af(x) + bf(y),
where a and b are arbitrary members of the field of scalars and x and y
and f(x) and f(y) are arbitrary members of suitable linear spaces.


While this is possible, and fairly trivial for a competent mathematician
to do, WM has not yet been able to do it.

But frequently claims already to have done it.
--

[Virgil]

In article
<68bd3b24-68ad-4d79...@x15g2000vbj.googlegroups.com>,

The issue here is whether some list can be so complete that no diagonal
can be constructed that is not already in it.

The answer, conceded above, is a resounding "NO"!

Which establishes, as required, that there is no surjection from the set
of all naturals to the set of all such listable sequences.

Thus WM concedes the proof of Cantor's diagonal argument.
--

[Virgil]

In article
<b3e95ba2-5b9a-4f05...@gp5g2000vbb.googlegroups.com>,

It clearly has not forced you, or even enticed you to make sense.
--

[Virgil]

In article
<e470cc8d-a195-4945...@w3g2000vba.googlegroups.com>,

One clock display suffices for infinitely many instants, two a day for
12 hour clocks.
--

[Virgil]

In article
<c89d12c1-637b-4262...@hq4g2000vbb.googlegroups.com>,

While the identity map on |R is an isomorphism, there is no clear way to
biject |R with the set of all binary sequences , especially if one has
to preserve linearity: f(b1 +b2) = f(b1) + f(b2) and f(r*b1) = r*f(b1),
for real r and binaries b1 and b2.
--

[Virgil]

In article
<81651b4b-e226-49c8...@g8g2000vbf.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:



> I am not too stupid to see s> It has been done already long ago (see Matheology � 226).

> 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".
--

[Virgil]

In article
<68bd3b24-68ad-4d79...@x15g2000vbj.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> 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,

For example:

[fom]

In pseudo-capitalist societies such as the United
States, merit is no guarantee for the advancement
of those born into families of little means.

That is the nature of free societies.

You should review Virgil's remarks.

He has suggested that you look up the
difference between a bijection and a linear
mapping.

It may be true that if one has x:=>x that
one also has a trivial isomorphism, but you have
not even demonstrated that you know what
else is required to satisfy the definitions.

And, for the record, I do not view a tree constructed
over an alphabet with two letters to be a real number.

Nor, if I allow you that much, can you use it that
way because of your finitism (as I explained to
Ross Finlayson).

I will not permit you to simply convert infinite
sequences to compact names so easily.

It is not *agreed* upon.

[Virgil]

In article
<b3e95ba2-5b9a-4f05...@gp5g2000vbb.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

Something seems to force WM to make an ass of himself with remarkable
frequency.

> 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.

There are several flaws in WM's claim that the identity map on |R
induces a linear map on {0,1}^|N, the set of all binary sequences.

[Virgil]

In article
<e470cc8d-a195-4945...@w3g2000vba.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

And a future whose end one cannot see.
>
> Regards, WM

######################################################################



WM has frequently claimed that HIS mapping from the set of all infinite
binary sequences to the set of paths of a CIBT is a linear mapping.

In order to show that such a mapping is a linear mapping, WM would first
have to show that the set of all binary sequences is a linear space
(which he has not done and apparently cannot do) and that the set of
paths of a CIBT is also a vector space (which he also has not done and
apparently cannot do) and then show that his mapping, say f, satisfies
the linearity requirement that f(ax + by) = af(x) + bf(y),
where a and b are arbitrary members of the field of scalars and x and y
and f(x) and f(y) are arbitrary members of suitable linear spaces.


While this is possible, and fairly trivial for a competent mathematician

to do.

I have done it but WM has not yet been able to do it.

[Virgil]

In article
<02b6c3ee-e9d2-45dc...@fn10g2000vbb.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 17 Mrz., 23:05, William Hughes <wpihug...@gmail.com> wrote:

> > On Mar 17, 10:49�pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > On 17 Mrz., 22:39, William Hughes <wpihug...@gmail.com> wrote:
> > > > You say that if a set of lines contains an unfindable
> > > > line it is necessary that there are
> > > > two findable lines.
> >
> > > No.
> >
> > Oh, so there can be a set of lines that contains an unfindable
> > line but not two findable lines ?!?
>
> When you remove every line as soon as you have found it, then no
> findable line remains. Isn't that obvious?

Why remove it so rapidly. Hold off at least until it has been proved
unneccessary by the finding of another line.

>
> However this might not be interesting for the majority of readers.
> Much more interesting will be how the case of actual infinity can be
> explained without contradicting the construction principle of our well-
> known list.

Such monstrosities as WM creates are only well known in
Wolkenmuekenheim.

> 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.

[Virgil]

In article
<9bc1333f-7b7b-4d39...@g8g2000vbf.googlegroups.com>,

Showing how one canfind an unfindable line?

Only on Wolkenmuekenheim.





=========================================================================

WM claimed:

[Virgil]

In article
<3df59ad9-d115-4db7...@k14g2000vbv.googlegroups.com>,

Works everywhere but in Wolkenmuekenheim.

[William Hughes]

On Mar 18, 4:45 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 18 Mrz., 13:50, William Hughes <wpihug...@gmail.com> wrote:
>
> > So you take a set of lines that contains an unfindable line
> > remove all the findable lines and end up with a set
> > that contains an unfindable line, but no findable line ?!?
>
> If you remove every findable line, there cannot remain a findable
> line, can it?

Nor can there remain an unfindable line.

>
> 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.

[WM]

On 19 Mrz., 00:44, William Hughes <wpihug...@gmail.com> 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?

Do you accept that a set of infinitely many lines contains at least
one subset of two lines?

Do you accept that every static (= existing in the sense of set
theory) set of lines has a (fixed and knowable) first line?

Regards, WM

[fom]

On 3/18/2013 10:45 AM, WM wrote:

> On 18 Mrz., 13:50, William Hughes <wpihug...@gmail.com> wrote:
>
>> So you take a set of lines that contains an unfindable line
>> remove all the findable lines and end up with a set
>> that contains an unfindable line, but no findable line ?!?
>
> If you remove every findable line, there cannot remain a findable
> line, can it?

Why don't you try *answering* what was asked.