Matheology § 223: AC and AMS

[WM]

Matheology § 223: AC and AMS

How obvious a contradiction has to result from an additional axiom in
order to reject it?

The Axiom of Choice (AC) states that every set can be well-ordered.

In order to well-order an uncountable set, an uncountable alphabet is
required, since a countable alphabet is not sufficient (compare the
Binary Tree: § 190). But an alphabet is a linearly ordered set
(otherwise you would never find most letters of the alphabet - compare
the telephone book). And linear ordering implies well-ordering.

So the Axiom of Choice contradicts the other ZF-axioms. (This has
already been shown by Hausdorff-Banach-Tarski who proved that by means
of AC we can prove that, after some turning and twisting, but without
any addition or subtraction of even one single point, the measurable
set V is identical with the measurable set 2V.)

With equal right we can introduce the Axiom of Meagre Sum (AMS)
stating: There is a set of n positive natural numbers with sum n*n/2.
This axiom is not constructive, since nobody can construct such a set.
But the disproof by the well known fact that the sum of n different
natural numbers is never less than n*(n+1)/2 is not less obvious than
the disproof of AC.

Regards, WM

[AMeiwes]

"WM" <muec...@rz.fh-augsburg.de> wrote in message
news:13e0e3df-9693-4243...@w3g2000vba.googlegroups.com...

>Matheology § 223: AC and AMS
>
>How obvious a contradiction has to result from an additional axiom in
>order to reject it?
>
>The Axiom of Choice (AC) states that every set can be well-ordered.

wrong, wiki says Axiom of Choice is "the product of a collection of
non-empty sets is non-empty"

>In order to well-order an uncountable set, an uncountable alphabet is
>required,

why use an alphabet ?
each term in an alphabet is represented by numbers.
so why use one?

>since a countable alphabet is not sufficient (compare the
>Binary Tree: § 190). But an alphabet is a linearly ordered set
>(otherwise you would never find most letters of the alphabet - compare
>the telephone book). And linear ordering implies well-ordering.

but none of this applies, countable set stuff, to your uncountable set
problem.

[WM]

On 9 Mrz., 19:14, "AMeiwes" <inva...@invalid.com> wrote:
> "WM" <mueck...@rz.fh-augsburg.de> wrote in message

>
> news:13e0e3df-9693-4243...@w3g2000vba.googlegroups.com...
>
> >Matheology § 223: AC and AMS
>
> >How obvious a contradiction has to result from an additional axiom in
> >order to reject it?
>
> >The Axiom of Choice (AC) states that every set can be well-ordered.
>
> wrong, wiki says Axiom of Choice  is "the product of a collection of
> non-empty sets is non-empty"

Zermelo created AC and titled his paper: Proof that every set can be
well-ordered.
>

> >In order to well-order an uncountable set, an uncountable alphabet is
> >required,
>
> why use an alphabet ?
> each term in an alphabet is represented by numbers.
> so why use one?

A set of numbers can also serve as the alphabet. Numbers or letters or
pictures or words: that is all the same. The detailled structure of
the "symbols" does not matter. In any case an uncountable (but
linearly ordered, hence countable) set of symbols is required.

Regards, WM

[Virgil]

In article
<13e0e3df-9693-4243...@w3g2000vba.googlegroups.com>,

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

> Matheology § 223: AC and AMS
>
> How obvious a contradiction has to result from an additional axiom in
> order to reject it?
>
> The Axiom of Choice (AC) states that every set can be well-ordered.

One is not required to assume the axiom of choice.

>
> In order to well-order an uncountable set, an uncountable alphabet is
> required, since a countable alphabet is not sufficient (compare the
> Binary Tree: § 190). But an alphabet is a linearly ordered set
> (otherwise you would never find most letters of the alphabet - compare
> the telephone book). And linear ordering implies well-ordering.

The reals are linearly ordered but not well ordered by that ordering,
and no explicit well ordering is known.

>
> So the Axiom of Choice contradicts the other ZF-axioms.

Actually, it has been proved independent of them, so that it cannot be
in contradiction with them.

> (This has
> already been shown by Hausdorff-Banach-Tarski who proved that by means
> of AC we can prove that, after some turning and twisting, but without
> any addition or subtraction of even one single point, the measurable
> set V is identical with the measurable set 2V.)

If you do not wish to accept the axiom of choice, no one can force you
to do so.

>
> With equal right we can introduce the Axiom of Meagre Sum (AMS)
> stating: There is a set of n positive natural numbers with sum n*n/2.

Axioms are not introduced for no reason, as your AMS would seem to be.

> This axiom is not constructive, since nobody can construct such a set.n

> But the disproof by the well known fact that the sum of n different
> natural numbers is never less than n*(n+1)/2 is not less obvious than
> the disproof of AC.

Then stick to ZF and scuttle C, if you wish, but only those who are
deeply into measure theory seem to have anything to be bothered about,
in assuming C.




And where is WM's proof that some mapping from the set of all binary
sequences to the set of all paths of a CIBT is a linear mapping?
WM several times claimed it but cannot seem to prove it.
--

[William Hughes]

On Mar 9, 10:56 am, WM <mueck...@rz.fh-augsburg.de> wrote:
<snip>

> This has
> already been shown by Hausdorff-Banach-Tarski who proved that by means
> of AC we can prove that, after some turning and twisting, but without
> any addition or subtraction of even one single point, the measurable
> set V is identical with the measurable set 2V.)
>

Well actually what they showed was that V and 2V can be
decomposed into the same set of unmeasureable sets.
However, I suppose in Wolkenmeukenheim this counts as
showing they are identical.

[Virgil]

In article <khfu77$ee8$1...@news.albasani.net>,

"AMeiwes" <inv...@invalid.com> wrote:

> "WM" <muec...@rz.fh-augsburg.de> wrote in message
> news:13e0e3df-9693-4243...@w3g2000vba.googlegroups.com...
> >Matheology § 223: AC and AMS

> >

> >The Axiom of Choice (AC) states that every set can be well-ordered.
>
> wrong, wiki says Axiom of Choice is "the product of a collection of
> non-empty sets is non-empty"

"THE" Axiom of Choice has a whole bunch of logically equivalent but
apparently quite disparate forms, and even disparate names, e.g., Zorn's
Lemma.
--

[Virgil]

In article <827cfcbf-c7fe-4e36...@googlegroups.com>,
harry_p...@walla.co.il wrote:

> On Saturday, March 9, 2013 11:56:56 AM UTC+2, WM wrote:
> > Matheology § 223: AC and AMS
> >
> >
> >
> > How obvious a contradiction has to result from an additional axiom in
> >
> > order to reject it?
> >
> >
> >
> > The Axiom of Choice (AC) states that every set can be well-ordered.
> >
> >
> >
> > In order to well-order an uncountable set, an uncountable alphabet is
> >
> > required, since a countable alphabet is not sufficient (compare the
> >
> > Binary Tree: § 190). But an alphabet is a linearly ordered set
> >
> > (otherwise you would never find most letters of the alphabet - compare
> >
> > the telephone book). And linear ordering implies well-ordering.
> >
> >
> >
> > So the Axiom of Choice contradicts the other ZF-axioms. (This has
> >
> > already been shown by Hausdorff-Banach-Tarski who proved that by means
> >
> > of AC we can prove that, after some turning and twisting, but without
> >
> > any addition or subtraction of even one single point, the measurable
> >
> > set V is identical with the measurable set 2V.)
> >
> >
> >
> > With equal right we can introduce the Axiom of Meagre Sum (AMS)
> >
> > stating: There is a set of n positive natural numbers with sum n*n/2.
> >
> > This axiom is not constructive, since nobody can construct such a set.
> >
>

> Poor WM: your desperation to write nonsenses must be driving you nuts: the
> set
>
> {0,1,2,5} with k = 4 natural element fulfills your ridiculous condition:
>
> 0 + 1 + 2 + 5 = 8 = 4*4/2 ...no additional axiom needed.
>
> You _really_ should go to Baden Baden and take some baths...but stay clear
> from
>
> gambling!

This one time WM was right, because in this case he specified POSITIVE
natural numbers, which excludes the 0 in your alleged counter-example.

Note that the 0 in your alleged counter-example is not a POSITIVE
natural number as required by WM's claim, so your alleged
counter-example fails to work as a counter-example.
--

[Virgil]

In article
<90272ca3-bad8-4d12...@o5g2000vbp.googlegroups.com>,

Linear ordering does not imply countability, at least it doesn't do so
outside of Wolkenmuekenheim.
>
For example, the set of reals is linearly ordered even inside
Wolkenmuekenheim, but, except in backwaters like Wolkenmuekenheim ,
uncountable.



And where is WM's long missing proof of his claim that some mapping from

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

WM several times claimed that such a LINEAR mapping exists but cannot
ever seem able to prove it.
--

[WM]

On 9 Mrz., 22:43, Virgil <vir...@ligriv.com> wrote:
> In article
> <13e0e3df-9693-4243-9a75-f4e5b33ee...@w3g2000vba.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > Matheology § 223: AC and AMS
>
> > How obvious a contradiction has to result from an additional axiom in
> > order to reject it?
>
> > The Axiom of Choice (AC) states that every set can be well-ordered.
>
> One is not required to assume the axiom of choice.
>

More, one is required to not accept it.

>
>
> > In order to well-order an uncountable set, an uncountable alphabet is
> > required, since a countable alphabet is not sufficient (compare the
> > Binary Tree: § 190). But an alphabet is a linearly ordered set
> > (otherwise you would never find most letters of the alphabet - compare
> > the telephone book). And linear ordering implies well-ordering.
>
> The reals are linearly ordered but not well ordered by that ordering,
> and no explicit well ordering is known.

The reals are not linearly ordered, otherwise you could determine
which real follows upon pi.

>
>
>
> > So the Axiom of Choice contradicts the other ZF-axioms.
>
> Actually, it has been proved independent of them, so that it cannot be
> in contradiction with them.

This proof shows that ZFC is inconsistent.

>
> > (This has
> > already been shown by Hausdorff-Banach-Tarski who proved that by means
> > of AC we can prove that, after some turning and twisting, but without
> > any addition or subtraction of even one single point, the measurable
> > set V is identical with the measurable set 2V.)
>

> > With equal right we can introduce the Axiom of Meagre Sum (AMS)
> > stating: There is a set of n positive natural numbers with sum n*n/2.
>
> Axioms are not introduced for no reason, as your AMS would seem to be.

The reason of AMS is to show with somewhat more contrast that set
theorists have drifted into inconstency during the last century.

Regards, WM

[WM]

On 10 Mrz., 00:14, William Hughes <wpihug...@gmail.com> wrote:
> On Mar 9, 10:56 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> <snip>
>
> > This has
> > already been shown by Hausdorff-Banach-Tarski who proved that by means
> > of AC we can prove that, after some turning and twisting, but without
> > any addition or subtraction of even one single point, the measurable
> > set V is identical with the measurable set 2V.)
>
> Well actually what they showed was that V and 2V can be
> decomposed into the same set of unmeasureable sets.

They showed that the set V, without adding one single point or
interval can be shown to have twice its volume (and more). Compare
Matheology § 087
http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf

But even more devastating to the trustworthiness of mathematics is the
fact that matheologians pretend that mathematicians have proved that
unlabelled labels are assumed to exist and to be capable of being well-
labelled

Regards, WM

[WM]

On 10 Mrz., 01:54, Virgil <vir...@ligriv.com> wrote:
> In article

> <90272ca3-bad8-4d12-9873-03843043c...@o5g2000vbp.googlegroups.com>,

>
>
>
>
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 9 Mrz., 19:14, "AMeiwes" <inva...@invalid.com> wrote:
> > > "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>
> > >news:13e0e3df-9693-4243...@w3g2000vba.googlegroups.com...
>
> > > >Matheology § 223: AC and AMS
>
> > > >How obvious a contradiction has to result from an additional axiom in
> > > >order to reject it?
>
> > > >The Axiom of Choice (AC) states that every set can be well-ordered.
>
> > > wrong, wiki says Axiom of Choice  is "the product of a collection of
> > > non-empty sets is non-empty"
>
> > Zermelo created AC and titled his paper: Proof that every set can be
> > well-ordered.
>
> > > >In order to well-order an uncountable set, an uncountable alphabet is
> > > >required,
>
> > > why use an alphabet ?
> > > each term in an alphabet is represented by numbers.
> > > so why use one?
>
> > A set of numbers can also serve as the alphabet. Numbers or letters or
> > pictures or words: that is all the same. The detailled structure of
> > the "symbols" does not matter. In any case an uncountable (but
> > linearly ordered, hence countable) set of symbols is required.
>
> Linear ordering does not imply countability,

That depends on the interpretation of linearity that has many meanings
in mathematics. Obviously an alphabet must be linearly ordered in the
sense that exactly one symbol follows upon another one.

Regards, WM

[William Hughes]

On Mar 10, 11:04 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 10 Mrz., 00:14, William Hughes <wpihug...@gmail.com> wrote:
>
> > On Mar 9, 10:56 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > <snip>
>
> > > This has
> > > already been shown by Hausdorff-Banach-Tarski who proved that by means
> > > of AC we can prove that, after some turning and twisting, but without
> > > any addition or subtraction of even one single point, the measurable
> > > set V is identical with the measurable set 2V.)
>
> > Well actually what they showed was that V and 2V can be
> > decomposed into the same set of unmeasureable sets.
>
> They showed that the set V, without adding one single point or
> interval can be shown to have twice its volume

Well I guess that in Wolkenmeukenheim you can determine
the volume of a set, X, by dividing it into unmeasureable
sets, recombining those unmeasurable sets into a measurable
set Y, and taking the volume of Y.

[WM]

No, I determine the volume of a set X, say the volume of a sphere, by
using the formula of Archimedes and in the same manner I determine the
volume of a set Y of two spheres.

There cannot be any discussion that the intermediate abracadabra is
irrelevant.

But as I already mentioned, the more alarming fact (with respect to
the mental health of matheologians, in particular such who proudly
boast to be logicans) are inventions like uncountable alphabets.

Regards, WM

[William Hughes]

On Mar 10, 11:51 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 10 Mrz., 11:39, William Hughes <wpihug...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On Mar 10, 11:04 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > On 10 Mrz., 00:14, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > On Mar 9, 10:56 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > <snip>
>
> > > > > This has
> > > > > already been shown by Hausdorff-Banach-Tarski who proved that by means
> > > > > of AC we can prove that, after some turning and twisting, but without
> > > > > any addition or subtraction of even one single point, the measurable
> > > > > set V is identical with the measurable set 2V.)
>
> > > > Well actually what they showed was that V and 2V can be
> > > > decomposed into the same set of unmeasureable sets.
>
> > > They showed that the set V, without adding one single point or
> > > interval can be shown to have twice its volume
>
> > Well I guess that in Wolkenmeukenheim you can determine
> > the volume of a set, X, by dividing it into unmeasureable
> > sets, recombining those unmeasurable sets into a measurable
> > set Y, and taking the volume of Y
>
> No, I determine the volume of a set X, say the volume of a sphere, by
> using the formula of Archimedes and in the same manner I determine the
> volume of a set Y of two spheres.
>
> There cannot be any discussion that the intermediate abracadabra is
> irrelevant.

The fact that the "intermediate abracadabra" is not volume
preserving is very relevant.

[WM]

It is point-preserving. Can you find any point that goes lost or is is
new? Can you understand volumes without points?
Therefore it is volume-preserving, or better, it would be volume-
preserving if it was meaningful at all.

Regards, WM

[William Hughes]

In Wolkenmuekenheim a procedure that is point preserving
is volume preserving.

[fom]

On 3/10/2013 6:55 AM, William Hughes wrote:
>
> In Wolkenmuekenheim a procedure that is point preserving
> is volume preserving.
>

Countably many points, don't forget that part.

[fom]

Nope.

Linear ordering does not imply countability.

One can have a linear order which is not dense.

And that is what you should say if that is what you mean.

[fom]

You assume, quite naturally, that any mathematical object you
require can be given a label. So there is little difference.

I agree with you concerning this problem. The role of semiotics
explicitly discussed by Bolzano and Frege disappeared because
of the influence of Russell, Wittgenstein, Carnap and Quine.
Your whining, however, is not how a mathematician should respond.

[fom]

Inconsistency is demonstrated by deriving a statement and its
negation from the axioms according to standard mathematical
arguments.

Everyone has been eager to see such a deduction on your part.

[WM]

On 10 Mrz., 12:55, William Hughes <wpihug...@gmail.com> wrote:
> In Wolkenmuekenheim a procedure that is point preserving
> is volume preserving.

And every number has a name. Unbelievable?

Regards, WM

[WM]

"Countable" is a nonsense notion of the kingdom of finished infinity.

Regards, WM

[WM]

What interpretations of linear do you know?
More than five?
Otherwise spare your comments.

>
> Linear ordering does not imply countability.

Linear ordering of an alphabet does.

>
> One can have a linear order which is not dense.
>
> And that is what you should say if that is what you mean.

I did not expect readers who missed to learn the alphabet and its
properties in school.

Regards, WM

[WM]

Numbers have names.
Numbers have no names.

Or:
Numbers without names cannot be well-ordered.
All numbers can be well-ordered.

Or:
An alphabet has the linear order of a sequence.
An uncountable alphabet cannot have that order.

That's enough.

> Everyone has been eager to see such a deduction on your part.-

Everyone who has not deliberately blinded himself can see.

Regards, WM

[WM]

On 10 Mrz., 13:13, fom <fomJ...@nyms.net> wrote:
> On 3/10/2013 5:04 AM, WM wrote:
>
>
>
>
>
> > On 10 Mrz., 00:14, William Hughes <wpihug...@gmail.com> wrote:
> >> On Mar 9, 10:56 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >> <snip>
>
> >>> This has
> >>> already been shown by Hausdorff-Banach-Tarski who proved that by means
> >>> of AC we can prove that, after some turning and twisting, but without
> >>> any addition or subtraction of even one single point, the measurable
> >>> set V is identical with the measurable set 2V.)
>
> >> Well actually what they showed was that V and 2V can be
> >> decomposed into the same set of unmeasureable sets.
>
> > They showed that the set V, without adding one single point or
> > interval can be shown to have twice its volume (and more). Compare
> > Matheology § 087
> >http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf
>
> > But even more devastating to the trustworthiness of mathematics is the
> > fact that matheologians pretend that mathematicians have proved that
> > unlabelled labels are assumed to exist and to be capable of being well-
> > labelled
>
> You assume, quite naturally, that any mathematical object you
> require can be given a label.  So there is little difference.

"Die Zahlen sind freie Schöpfungen des menschlichen Geistes, sie
dienen als ein Mittel, um die Verschiedenheit der Dinge leichter und
schärfer aufzufassen." Numbers are created to easen perception of
reality. (Dedekind)

Numbers without names cannot have been created, cannot help to
perceive anything (except the stupidity of matheology) and cannot be
well-ordered themselved, because ordering of immaterial thnings cannot
hapen without defining these things.

>
> I agree with you concerning this problem.

Fine.

>  The role of semiotics

Nothing of that sort is required - only a healthy and sober brain.

Regards, WM

[fom]

This is your first incorrect statement that *might* be
forgiven because of context:

"but linearly ordered, hence countable"

This is, by virtue of standard definitions for
"linear order," your outright lie:

"That depends on the interpretation of linearity that has many
meanings in mathematics."

And, this is not obvious at all:

"Obviously an alphabet must be linearly ordered in the
sense that exactly one symbol follows upon another one."

One may grant that the semiotics of naming objects in a
logical system requires a well-ordered introduction of
canonical names. But, that is different from claims that
some arbitrary collection of symbols that serve as an
"alphabet" is necessarily ordered at all.

You should learn to be precise rather than persist with
your lazy, indolent habits. You are as bad as any
"matheologist" -- worse, in fact.

And just so we are clear about the nature of order
with regard to an "alphabet", one has

"Alphabets were defined above as collections of
symbols, which we agree to regard as letters. It
is convenient to depict such a collection as a
word composed of the alphabet's letters, taken
in a definite order, for example, in the order of
their succession. In what follows, we shall often
act in precisely this fashion. Nevertheless, the
order of the letters in an alphabet will not, as
a rule, be of interest to us. Bearing this in mind,
we shall call two alphabets consisting of the same
letters equicomposed, and all our investigations
involving alphabets will be carried out up to an
equicomposition of the alphabets figuring in
them."

The Theory of Algorithms
A.A. Markov



So, whereas the assignment of symbols as names
for objects clearly has a natural association with
well ordering, the mere fact of being a linguistic
symbol does not.

[gus gassmann]

On 10/03/2013 9:19 AM, fom wrote:
> On 3/10/2013 4:55 AM, WM wrote:
>> On 9 Mrz., 22:43, Virgil <vir...@ligriv.com> wrote:
>>> In article
>>> <13e0e3df-9693-4243-9a75-f4e5b33ee...@w3g2000vba.googlegroups.com>,
>>>
>>> WM <mueck...@rz.fh-augsburg.de> wrote:
>>>> Matheology � 223: AC and AMS
>>>
>>>> How obvious a contradiction has to result from an additional axiom in
>>>> order to reject it?
>>>
>>>> The Axiom of Choice (AC) states that every set can be well-ordered.
>>>
>>> One is not required to assume the axiom of choice.
>>>
>> More, one is required to not accept it.
>>>
>>>
>>>> In order to well-order an uncountable set, an uncountable alphabet is
>>>> required, since a countable alphabet is not sufficient (compare the
>>>> Binary Tree: � 190). But an alphabet is a linearly ordered set
>>>> (otherwise you would never find most letters of the alphabet - compare
>>>> the telephone book). And linear ordering implies well-ordering.
>>>
>>> The reals are linearly ordered but not well ordered by that ordering,
>>> and no explicit well ordering is known.
>>
>> The reals are not linearly ordered, otherwise you could determine
>> which real follows upon pi.

Here the Great And Illustrious Professor of Mathematics Wolfgang
Mueckenheim of the Glorious University of Applied Sciences in Augsburg,
Germany, demonstrates once again (surprise, surprise) his total
ignorance of and disregard for mathematical terminology.

A linear order (or total order to distinguish it from a partial order)
requires no more than trichotomy. The real r which "follows upon pi"
would be the smallest element in the set {x: x > pi}. In order to
guarantee that such an r exists one needs a _well-ordering_ on the reals
(which is equivalent to AC). This has been explained to the good
professor a good many times before, but he is evidently too dense to
even understand mathematical definitions.

[WM]

On 10 Mrz., 14:27, fom <fomJ...@nyms.net> wrote:

> "Obviously an alphabet must be linearly ordered in the
> sense that exactly one symbol follows upon another one."
>
> One may grant that the semiotics of naming objects in a
> logical system requires a well-ordered introduction of
> canonical names.

Well-ordering (every non-empty subset has a first element) is not
enough. You must be able to find the position of every symbol. This
requires lexical ordering. (No system can learn the position of more
symbols than its memory allows.)

>  But, that is different from claims that
> some arbitrary collection of symbols that serve as an
> "alphabet" is necessarily ordered at all.

How should an alphabet be used if it is not ordered and the symbols
cannot be found?

> "Alphabets were defined above as collections of
> symbols, which we agree to regard as letters.  It
> is convenient to depict such a collection as a
> word composed of the alphabet's letters, taken
> in a definite order, for example, in the order of
> their succession.

Order seems to be important?

> In what follows, we shall often
> act in precisely this fashion.  Nevertheless, the
> order of the letters in an alphabet will not, as
> a rule, be of interest to us.

Perhaps the order may be of no interest if there are other means to
transfer a selected symbol into the CPU.

In any case, if a language shall be speakable, it is required that the
symbols can be selected and transferred without long seeking. That
implies a lexical or related linear ordering of symbols, if there are
too many to be learned by heart.

For languages that shall not be applied, except in matheology, this
may be different. But the latter is as irrelevant for healthy minds as
is most contents of matheology.

Regards, WM

[William Hughes]

On Mar 10, 1:33 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 10 Mrz., 12:55, William Hughes <wpihug...@gmail.com> wrote:
>
> > In Wolkenmuekenheim a procedure that is point preserving
> > is volume preserving.
>

Question: Let f be a bijection. Is f point preserving?

[Virgil]

In article
<fe507fb2-e10c-466a...@h9g2000vbk.googlegroups.com>,

for WM's information:

A linear mapping in mathematics must be a special sort of mapping from
one linear space (vector space) to another.

Notation: O is the zero of field F and o is the zero element of group G.
Given field (F,+,*,O,1) and additive group (G,Ý,o),

G can be a linear space over F ONLY IF all of the following hold:
1. for each f in F and g in G, f€g is a member of G
2. For O and 1 in F and any g in G, O€g = o and 1€g = g
3. for f1 and f2 in F and g in G, (f1€g) Ý (f2€g) = (f1 + f2)€g
and (f1*f2)€g=f1€(f2€g)
4. for f in F and g1,g2 in G, f€(g1 Ý g2)=(f€g1) Ý (f€g2)

And a mapping, M, from one linear space to another will be a linear
mapping only if

M(f€(g1 Ý g2) = f( M(g1) Ý M(g2)) and M((f1 + f2)€g) = (f1 + f2)€M(g)
--

[Virgil]

In article
<5e27321f-996c-4ccc...@k14g2000vbv.googlegroups.com>,

Only if one requires a finite alphabet.

> >
> > One can have a linear order which is not dense.
> >
> > And that is what you should say if that is what you mean.
>
> I did not expect readers who missed to learn the alphabet and its
> properties in school.

And we do not expect those who apparently totally missed out on geometry
in school, like WM apparently did, to claim to be mathematicians.

Since some of the most common linear orders in real mathematics are NOT
finite, much less discrete, as in geometry, why should anyone assume
that one intends to limit them in such an artificial way?
>
> Regards, WM
--

[AMeiwes]

"WM" <muec...@rz.fh-augsburg.de> wrote in message
news:5e27321f-996c-4ccc...@k14g2000vbv.googlegroups.com...

On 10 Mrz., 13:06, fom <fomJ...@nyms.net> wrote:
> On 3/10/2013 5:07 AM, WM wrote:
>
>
>

<snipped>

>Otherwise spare your comments.
>
> Linear ordering does not imply countability.

if the set has complex elements, how do you create linear ordering ?



>Regards, WM

[Virgil]

In article
<3f202467-dca1-44d4...@z3g2000vbg.googlegroups.com>,

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

> On 10 Mrz., 14:27, fom <fomJ...@nyms.net> wrote:
>
> > "Obviously an alphabet must be linearly ordered in the
> > sense that exactly one symbol follows upon another one."
> >
> > One may grant that the semiotics of naming objects in a
> > logical system requires a well-ordered introduction of
> > canonical names.
>
> Well-ordering (every non-empty subset has a first element) is not
> enough. You must be able to find the position of every symbol. This
> requires lexical ordering. (No system can learn the position of more
> symbols than its memory allows.)

Lexical ordering of words, unless there is a fixed limit on world
length, will allow infinitely many words having no immediate
predecessors. Consider the problem of lexically ordering all finite
binary sequences with no limit on their finite lengths.

>
> >  But, that is different from claims that
> > some arbitrary collection of symbols that serve as an
> > "alphabet" is necessarily ordered at all.
>
> How should an alphabet be used if it is not ordered and the symbols
> cannot be found?

Consider written Chinese or Japanese, which have no alphabet of the sort
you demand.

>
> > "Alphabets were defined above as collections of
> > symbols, which we agree to regard as letters.  It
> > is convenient to depict such a collection as a
> > word composed of the alphabet's letters, taken
> > in a definite order, for example, in the order of
> > their succession.
>
> Order seems to be important?
>
>
> > In what follows, we shall often
> > act in precisely this fashion.  Nevertheless, the
> > order of the letters in an alphabet will not, as
> > a rule, be of interest to us.
>
> Perhaps the order may be of no interest if there are other means to
> transfer a selected symbol into the CPU.

The Chinese seem to manage it without an alphabet at all.

>
> In any case, if a language shall be speakable, it is required that the
> symbols can be selected and transferred without long seeking. That
> implies a lexical or related linear ordering of symbols, if there are
> too many to be learned by heart.

It is not required that a language have any writable symbols at all in
oder to be spoken, and nobody seems to learn their first language from
written symbols, though they may thereafter manage learn others from
such written symbols

>
> For languages that shall not be applied, except in matheology, this
> may be different. But the latter is as irrelevant for healthy minds as
> is most contents of matheology.

So WM denies that Chinese or Japanese languages are capable of conveying
mathematics?

WM seems to be willing to go to any lengths of nonsensibleness to
protect his corrupt world views.


-------------------


WM has 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
f(ax + by) = af(x) + bf(y), where a and b are arbitrary members of the
field of scalars and x and y are binary sequences and f(x) and f(y) are
paths in a CIBT.

By the way, WM, what are ax and 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.
--

[Virgil]

In article
<dec55044-da91-480f...@r9g2000vbh.googlegroups.com>,

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

> On 10 Mrz., 00:14, William Hughes <wpihug...@gmail.com> wrote:
> > On Mar 9, 10:56 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > <snip>
> >
> > > This has
> > > already been shown by Hausdorff-Banach-Tarski who proved that by means
> > > of AC we can prove that, after some turning and twisting, but without
> > > any addition or subtraction of even one single point, the measurable
> > > set V is identical with the measurable set 2V.)
> >
> > Well actually what they showed was that V and 2V can be
> > decomposed into the same set of unmeasureable sets.
>
> They showed that the set V, without adding one single point or
> interval can be shown to have twice its volume (and more). Compare
> Matheology § 087

> http://www.hs-augsburg.de/~mueckenh/KB/WMytheology.pdf

What was proved was that two apparently congruent sets have different
volumes.

>
> But even more devastating to the trustworthiness of mathematics is the
> fact that matheologians pretend that mathematicians have proved that
> unlabelled labels are assumed to exist and to be capable of being well-
> labelled

None of which is as devastating to mathematics s would be WM's claims
without proofs, if anyone other than himself was to be persuaded by them.
--

[Virgil]

In article
<c8cc4949-4c48-4715...@he10g2000vbb.googlegroups.com>,

A requirement that WM only deludes himself that he possesses.

For example consider his unproven claim of linearity:

WM has 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
f(ax + by) = af(x) + bf(y), where a and b are arbitrary members of the
field of scalars and x and y are binary sequences and f(x) and f(y) are
paths in a CIBT.

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

[Virgil]

In article
<2f305841-f2a4-4e5c...@fn10g2000vbb.googlegroups.com>,

If WM's definition or alphabets, or words, are such that only finitely
many can exist, fine, but he has not the power to make others accept
those definitions, and, as we can see, he is not actually persuading
anyone to accept them.

As a philosopher WM has great admiration for himself, but as a
mathematician he is seen by every one to suck.

WM has 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

f(ax + by) = af(x) + bf(y), where a and b are arbitrary members of a

field of scalars and x and y are binary sequences and f(x) and f(y) are
paths in a CIBT.

By the way, WM, what are ax and by and ax+by when x and y are binary

[Virgil]

In article
<344efb4d-e221-40f9...@x15g2000vbj.googlegroups.com>,

The set of points in a line can be bijected with the set of points in a
plane or even a real space of countably infinite dimension.

So according to WM, they should all have the same "volume"?

> Therefore it is volume-preserving, or better, it would be volume-
> preserving if it was meaningful at all.

Then, at least according to WM, bijections between a line segment and a
solid cube must be equally volume-preserving.

WM has 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

f(ax + by) = af(x) + bf(y), where a and b are arbitrary members of the

[Virgil]

In article
<3641dca8-9eb3-48e0...@o5g2000vbp.googlegroups.com>,

Since there are procedures that are point preserving between any two
cubes, regardless of their edge length, it follows that in
Wolkenmuekenheim all cubes are of equal volume.

WM has also claimed that a mapping from the set of all infinite binary

[Virgil]

In article
<855f7759-fb39-4d71...@c10g2000vbt.googlegroups.com>,

It is only thought to be nonsense in the kingdom of Wolkenmuekenheim
whose king and dictator is its only lonely resident.

Outside Wolkenmuekenheim one can distinguish between certain sets that
are provably countable and others which are provably not.

Within Wolkenmuekenheim it is required that there always be a mythical,
magical and unfindable natural number which does not have any successor.
--

[Virgil]

In article
<c4a85937-a7e1-4686...@w3g2000vba.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> On 9 Mrz., 22:43, Virgil <vir...@ligriv.com> wrote:
> > In article

> > <13e0e3df-9693-4243-9a75-f4e5b33ee...@w3g2000vba.googlegroups.com>,
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:

> > > Matheology § 223: AC and AMS
> >
> > > How obvious a contradiction has to result from an additional axiom in
> > > order to reject it?
> >
> > > The Axiom of Choice (AC) states that every set can be well-ordered.
> >

> > One is not required to assume the axiom of choice.
> >

> More, one is required to not accept it.

Only in the kingdom of Wolkenmuekenheim, outside of which WM has no
power to require anything of anyone. Except maybe his poor students.

> >
> >
> > > In order to well-order an uncountable set, an uncountable alphabet is

> > > required, since a countable alphabet is not sufficient (compare the
> > > Binary Tree: § 190). But an alphabet is a linearly ordered set
> > > (otherwise you would never find most letters of the alphabet - compare
> > > the telephone book). And linear ordering implies well-ordering.
> >
> > The reals are linearly ordered but not well ordered by that ordering,
> > and no explicit well ordering is known.
>
> The reals are not linearly ordered, otherwise you could determine
> which real follows upon pi.

A linear ordering does not require that any member have a "next larger"
or "next smaller", it merely requires that of any two distinct members,
one of them comes before the other in that ordering.

What WM may be trying to speak about is a sequential ordering.

> >
> >
> >
> > > So the Axiom of Choice contradicts the other ZF-axioms.
> >
> > Actually, it has been proved independent of them, so that it cannot be
> > in contradiction with them.
>
> This proof shows that ZFC is inconsistent.

What WM claims WM never manages to prove.

Lets see WM try to prove that ZFC is inconsistent.
> >
> > > (This has

> > > already been shown by Hausdorff-Banach-Tarski who proved that by means
> > > of AC we can prove that, after some turning and twisting, but without
> > > any addition or subtraction of even one single point, the measurable
> > > set V is identical with the measurable set 2V.)
> >

> > > With equal right we can introduce the Axiom of Meagre Sum (AMS)
> > > stating: There is a set of n positive natural numbers with sum n*n/2.
> >
> > Axioms are not introduced for no reason, as your AMS would seem to be.
>
> The reason of AMS is to show with somewhat more contrast that set
> theorists have drifted into inconstency during the last century.

It only shows that WM has drifted into inconsistency.

As in WM's inability to use "linear mapping" correctly.

A proper mathematician after claiming that a particular mapping was a
linear mapping and being challenged, would either produce a proof of
that claim or withdraw the claim.

WM does neither.

WM has claimed several time that some unspecified mapping from the set
of all binary sequences to the set of all paths of a CIBT is a linear
mapping.

I have challenged him either to prove that claim or withdraw it.

So far he has not done either, but merely repeated the claim.

That is not the behavior of a proper mathematician.

So I again challenge him either to prove it or withdraw it.
--

[Virgil]

In article
<1035fb16-17c6-46db...@u2g2000vbx.googlegroups.com>,

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


> Everyone who has not deliberately blinded himself can see.
>

I am not blind but yet I do not see either WM's proof of his claim to
have a linear mapping from the set of all binary sequences to the set of
all paths of a CIBT or his retraction of that claim..
--

[WM]

Question: Can there exist a volume without any point in it?

Regards, WM

[fom]

You have previously stated that in your reality
you cannot distinguish between the idea of
a number and the idea of a name.

Standard mathematical argument does not admit
self-contradictory statements into arguments.

>
> Or:
> Numbers without names cannot be well-ordered.
> All numbers can be well-ordered.
>

You have previously denied the existence of
numbers without names.

Standard mathematical argument does not admit
reference to objects that have been previously
denied existential import.

> Or:
> An alphabet has the linear order of a sequence.
> An uncountable alphabet cannot have that order.
>

I have already shown an example of a definition
for an alphabet from a cited source that rejects
the notion of an alphabet as having any intrinsic
order.

You need to define "alphabet".

You need to define "uncountable alphabet" as a
species of alphabet.

Then you need to explain how your definition
supports

"each alphabet has *the linear order of a sequence*"

"some species of alphabet do not have *the linear order of a sequence*"

This is how adjectives work in the classical logic
of genera and species.


> That's enough.
>

"That" being nothing.

Your claims require you to use the axioms of set
theory (any commonly accepted version) and argue
according to methods that are generally recognized
to be representable in the deductive calculus of
first-order logic.

>> Everyone has been eager to see such a deduction on your part.-
>
> Everyone who has not deliberately blinded himself can see.

As can anyone who has deliberately educated themselves.

[WM]

On 10 Mrz., 23:32, Virgil <vir...@ligriv.com> wrote:

>
> > > Linear ordering does not imply countability.
>
> > Linear ordering of an alphabet does.
>
> Only if one requires a finite alphabet.

No.

> Since some of the most common linear orders in real mathematics are NOT
> finite, much less discrete,

Any usable alphabet is discrete. Unusable alphabets are not part of
mathematics (neither linguistic nor any meaningful system outside of
matheology).

Regards, WM

[WM]

On 10 Mrz., 23:42, "AMeiwes" <inva...@invalid.com> wrote:
> "WM" <mueck...@rz.fh-augsburg.de> wrote in message

If the complex elements are discrete, one could use Cantor's zigzag
method.

But it is not at all the question how to create a linear ordering. An
alphabet, by definition and by practical application, is not an
alphabet unless it is discrete and linearly ordered, i.e., unless it
is a sequence and, therefore, countable.

Regards, WM

[WM]

On 10 Mrz., 23:50, Virgil <vir...@ligriv.com> wrote:

> Lexical ordering of words, unless there is a fixed limit on world
> length, will allow infinitely many words having no immediate
> predecessors. Consider the problem of lexically ordering all finite
> binary sequences with no limit on their finite lengths.
>

In an alphabet, however, every symbol except the first one, will have
a predecessor, and every symbol, except the last one, will have a
successor. Otherwise it cannot be applied and is not an applicable
alphabet.

Perhaps I should have said: I talk about usable alphabets for
speakable languages (in case a matheologian might introduce his
unusable alphabets for his unspeakable languages).

> > How should an alphabet be used if it is not ordered and the symbols
> > cannot be found?
>
> Consider written Chinese or Japanese, which have no alphabet of the sort
> you demand.

It is impossible to find a certain symbol other than searching all
symbols?
Perhaps that's why Chines is so difficult to learn. I don't know. But
anyhow it is possible to put all symbols in a given sequence if there
is at least one written catalogue of all symbols.

Regards, WM

[WM]

On 10 Mrz., 23:58, Virgil <vir...@ligriv.com> wrote:

> 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)?

ax is the binary sequence belonging to ax. What else should it be?
If a is larger than 1, some binary sequences do not exist in the
considered unit interval.

If f(x) is a path in the CIBT, then af(x) is the path belonging to
af(x).

>
> Until these and a few other issues are settled,

What is there to settle?

Take, for the sake of simplicity, paths of the decimal tree and
decimal sequences of the unit interval. Calculate exactly as you are
used to do.
If x = 0.1000... and a = 0.5000..., then ax = 0.05000...
If f(x) = 0.1000... and a = 0.5000..., then af(x) = 0.05000...
If desired convert into binaries.

If you leave the unit interval, then the results are not belonging to
the unit interval and not belonging to the tree.

Regards, WM

[WM]

On 11 Mrz., 00:12, Virgil <vir...@ligriv.com> wrote:

> The set of points in a line can be bijected with the set of points in a
> plane or even a real space of countably infinite dimension.

In geometry there is no volume without points. If the bijection-method
can be used to show that V = 2V, then the bijection-method is in
contradiction with mathematics. More is not to say.

Regards, WM

[WM]

On 11 Mrz., 09:17, fom <fomJ...@nyms.net> wrote:

>
> You have previously stated that in your reality
> you cannot distinguish between the idea of
> a number and the idea of a name.

That is not correctly quoted.
Numbers are names, but not all names are numbers.

>
> > Or:
> > Numbers without names cannot be well-ordered.
> > All numbers can be well-ordered.
>
> You have previously denied the existence of
> numbers without names.

Of course, but here I talk about matheology which accepts and requires
numbers without names. Did you really miss that?

> Your claims require you to use the axioms of set
> theory (any commonly accepted version) and argue
> according to methods that are generally recognized
> to be representable in the deductive calculus of
> first-order logic.

Absolutely wrong. My claim is that a language is speakable, an
alphabet is usable and a number is nameable. Any theory not obeying
these facts is nonsense. There is no need to analyze its details.
Wasted time.

Regards, WM

[fom]

On 3/10/2013 10:11 AM, WM wrote:
>
> How should an alphabet be used if it is not ordered and the symbols
> cannot be found?

Sadly, the foundational investigations of mathematics really
mangles the jargon of other disciplines. More or less, we grunt
at one another. So, whatever "language" is, it begins with aural
phonemes. As language develops "grammatical" structure, it is
organized into morphemes. One is then able to analyze language
and discern "grammatical knowledge" as such is referred to in my
copies of Aristotle.

By the time one gets to a discussion of "alphabets" one is
essentially faced with Lewis Caroll's nursery rhyme about
Humpty Dumpty. It is analysis that cannot justify its
own synthesis ("all the king's horse and all the king's
men couldn't put Humpty together again").

It is also, however, where mathematicians find themselves
since whatever "mathematical objects" might actually be,
their abstract nature and the ever present skeptical
criticism associated with them will inevitably
lead to consideration of how to organize inherently
meaningless syntax.

These remarks should not be construed as denying the
legitimacy of that mathematics purposely constructed
from alphabets.

Since my own views accept Frege's rejection of logicism, I
can offer you an answer to your question.

In Mach's rejection of the Kantian characterization of the
a priority of space, he asserts that there is a simultaneity
with respect to the relations of space and the situation of
objects within space. Along those lines, you will find an
introduction of "logical constants" organized into a
projective plane at

news://news.giganews.com:119/Jr2dnbdYvtfPdlrN...@giganews.com

Although I used Aristotelian terms for 4 of the elements
beyond the names corresponding to the 16 Boolean valued
functions, it is of no consequence to view the extension
to the Boolean names as being any four symbols similarly
related by unary negations. So,

ALL -> Ax
SOME -> Ex
NO -> Ax-
OTHER-> Ex-

In this, the unary negation must be within the scope
of the quantifier so that the complex is meaningful
with respect to its geometric situation and not by
virtue of the syntactic role given to the sign of
negation. That syntactic role is artificial because
unary negation is eliminable.

The use of geometry here denies independent meaning of
the unary negation as a logical connective. The cost
of that denial is the treatment of "logical constants"
as a system that is taken to be presupposed by any
analysis of its individual parts.

This had not been available to Frege or Russell when
they pursued their logical studies. To my knowledge,
Wittgenstein introduced truth tables, although it may
have been Post.

Along similar lines, I am generally unconcerned with
"parts" of deductions. If one is going to perform
deductions formally, one begins with sentences (that is,
fully quantified statements that are either true or
false) and one ends with sentences. So, "models" in
the sense of Tarski are not treating of a demonstrative
science in the sense of Aristotle.

If one then understands that "names" are (in principle)
introduced into mathematics by description, then those
names are introduced sequentially relative to the
Aristotelian sense of "that which is prior and that
which is posterior" inside of a deduction.

The general ideas are at

news://news.giganews.com:119/5bidnemPpsnq13zN...@giganews.com


but it is long and unwieldly. So do not really
waste your time.

However, if I have my logical constants, well-formed
formulas, and transformation rules, then every symbol
different from a logical constant or variable
is presumed to be obtained by description in an
order grounded by the fact that names are introduced
one at a time.

This illustrative example is excerpted from the
post mentioned above:

> v_0 EQ v_0
> |Name(V_0)=V()
> |v_1 EQ v_1
> ||-(v_1 = v_0)
> |||Name(v_1)=null()
> |||v_2 EQ v_2
> ||||((-(v_2 = v_0)) /\ (-(v_2 = v_1)))
> |||||Set(v_2)
> ||||||(null() c v_2)
> ||||||Set(v_1)
> |||||||Name(V_2)=R
> |||||||v_3 EQ v_3
> ||||||||(((-(v_3 = v_0)) /\ (-(v_3 = v_1))) /\ (-(v_3 = v_2)))
> |||||||||Set(v_1)
> ||||||||||Name(V_3)=L


Unlike you, I am ambivalent concerning infinities except
that I accept that there is a mathematical need for infinity
in relation to certain aspects associated with definition
described most clearly by Aristotle. Its imposition
into mathematics, however, has to do with Leibniz laws.
These opinions are not standard.

Nevertheless, I am realistic concerning the difficulties
imposed by infinities. In the conclusion of the post from
which that illustration is taken, you will find the
remarks:


> There is one thing that is conspicuously
> missing from all foundational literature.
> It is something that Skolem recognized,
> although his remarks were intended to
> be critical:
>
>
> "In order to obtain something absolutely
> non-denumerable, we would have to have
> either an absolutely nondenumerable
> number of axioms or an axiom that could
> yield an absolutely nondenumerable
> number of first-order axioms."
>
>
>
> What Skolem is describing is exactly what
> Descartes did when he placed numbers on
> the field of geometric points. There is
> a presupposition that the number system
> constitutes a consistent global labelling
> of the geometric individuals. For over
> a century, untenable philosophical
> positions involving arithmetization
> and logicism have obscured the simplest
> of historical facts.


What this would mean is that every purported
model of set theory would be required to
demonstrate the existence of a canonical
well-ordering of the model in order to
be taken as faithfully representing a
system in which "names" carry existential
import.

Well, if you have bothered to read this
far, you can see that I have no need of
an "alphabet" that is intrinsically
ordered. You may disagree, but I see
no value in driving the analysis to a
point where synthesis is impossible.
That is exactly how I view "alphabets".

On the other hand, there can be a need
for a canonical enumeration,

news://news.giganews.com:119/AuqdnYcXm8eaLVzN...@giganews.com



Anyway, you asked. I have given you an answer. It is
unlikely to be anything you will appreciate.

[WM]

On 11 Mrz., 11:32, fom <fomJ...@nyms.net> wrote:


> If one then understands that "names" are (in principle)
> introduced into mathematics by description, then those
> names are introduced sequentially relative to the
> Aristotelian sense of "that which is prior and that
> which is posterior" inside of a deduction.

Of course every name has to be explaned by other names or by showing
the objects. In principle all senses including nose and ears are
capable of perceiving definitions. It is an unhealthy state, induced
by people who need cruches for thinking, that there should be only one
written language called ZFC + FOPL.

Every idea that can be transferred and every carrier of this idea is
capable of belonging to science. There is not the least reason to
believe that a formal proof in written language is preferable to
spoken words or to drawings or pictures.

>
> The general ideas are at
>

> news://news.giganews.com:119/5bidnemPpsnq13zNnZ2dnUVZ_sOdn...@giganews.com

Skolem is right. In another text he criticizes Hilbert's missing
understanding of the relativity of uncountability.

But whether or not uncountable sets are introduced by an axiom like my
ASM or "Wiles' proof can be circumvented" or other nonsensical claims:
Everyone with minimal intelligence should be able to recognize that
things which have no material existence cannot be ordered or well-
ordered if there are no labels to label them.

> > What Skolem is describing is exactly what
> > Descartes did when he placed numbers on
> > the field of geometric points. There is
> > a presupposition that the number system
> > constitutes a consistent global labelling
> > of the geometric individuals. For over
> > a century, untenable philosophical
> > positions involving arithmetization
> > and logicism have obscured the simplest
> > of historical facts.
>
> What this would mean is that every purported
> model of set theory would be required to
> demonstrate the existence of a canonical
> well-ordering of the model in order to
> be taken as faithfully representing a
> system in which "names" carry existential
> import.
>
> Well, if you have bothered to read this
> far,

In fact, I did - with interest.

> you can see that I have no need of
> an "alphabet" that is intrinsically
> ordered. You may disagree, but I see
> no value in driving the analysis to a
> point where synthesis is impossible.

If you want to form words you are in need of letters. If you cannot
find them, you cannot use them. An alphabet without order is useless.

Regards, WM

[AMeiwes]

"WM" <muec...@rz.fh-augsburg.de> wrote in message

news:24048720-e6a3-4810...@x15g2000vbj.googlegroups.com...

Wrong. Both Chinese and Japanese alphabets cannot be put in linear order.

>
> Regards, WM

[AMeiwes]

"WM" <muec...@rz.fh-augsburg.de> wrote in message

news:<49740516-cc2b-4128...@w3g2000vba.googlegroups.com>...

> On 10 Mrz., 23:32, Virgil <vir...@ligriv.com> wrote:
>
> >
> > > > Linear ordering does not imply countability.
> >
> > > Linear ordering of an alphabet does.
> >
> > Only if one requires a finite alphabet.
>
> No.
>
> > Since some of the most common linear orders in real mathematics are NOT
> > finite, much less discrete,
>
> Any usable alphabet is discrete.

not so, in both chinese and japanese there are characters that represent
partial other characters so there is not a one to one mapping.

>Unusable alphabets are not part of
> mathematics

i.e.... "...unusable mathematics are not part of alphabets" .

>(neither linguistic nor any meaningful system outside of
> matheology).

"....any meaningful system outside of mathology" can be geology, math,
spamorama, zippy the pinhead, microbs, etc.....


>
> Regards, WM

[WM]

On 11 Mrz., 17:18, "AMeiwes" <inva...@invalid.com> wrote:
> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>
> news:24048720-e6a3-4810...@x15g2000vbj.googlegroups.com...
>
>
>
>
>
> > On 10 Mrz., 23:42, "AMeiwes" <inva...@invalid.com> wrote:
> >> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>
> >>news:5e27321f-996c-4ccc...@k14g2000vbv.googlegroups.com...
> >> On 10 Mrz., 13:06, fom <fomJ...@nyms.net> wrote:> On 3/10/2013 5:07 AM,
> >> WM wrote:
>
> >> <snipped>
>
> >> >Otherwise spare your comments.
>
> >> > Linear ordering does not imply countability.
>
> >> if the set has complex elements, how do you create linear ordering ?
>
> > If the complex elements are discrete, one could use Cantor's zigzag
> > method.
>
> > But it is not at all the question how to create a linear ordering. An
> > alphabet, by definition and by practical application, is not an
> > alphabet unless it is discrete and linearly ordered, i.e., unless it
> > is a sequence and, therefore, countable.
>
> Wrong.   Both Chinese and Japanese alphabets cannot be put in linear order.

That means there are more symbols than rational numbers (which can be
put in linear order)?
How could dictionaries work?
Who told you so?
And why do you believe it?

Look: There are less than 10^100 Chinese charaters. You may inform
yourself here:
http://en.wikipedia.org/wiki/Chinese_characters

Regards, WM

[AMeiwes]

"WM" <muec...@rz.fh-augsburg.de> wrote in message

news:5771bf4f-1f15-4c96...@14g2000vbr.googlegroups.com...

On 11 Mrz., 17:18, "AMeiwes" <inva...@invalid.com> wrote:
> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>
> news:24048720-e6a3-4810...@x15g2000vbj.googlegroups.com...
>
>
>
>
>

>> Wrong. Both Chinese and Japanese alphabets cannot be put in linear order.

>That means there are more symbols than rational numbers (which can be
>put in linear order)?

wrong concusion again.
numbers are sequential, and therefore can be put in linear order.

If your element of your set are multivalued like complex numbers, or
tri-numbers, or multivalued, or multi-meaning like alphabets, they cannot be
put in Linear Order.

Chinese and Japaniese characters are logograms, some with multiple meanings
or contextual, but they are not sequential, and are not put in linear order.

>How could dictionaries work?

http://www.saiga-jp.com/kanji_dictionary.html now put those in "Liner
Order"

>Who told you so?

Who told you that all alpabets can be put in linear order, and are therefore
have less symboles than rational numbers ?

>And why do you believe it?

it is obvious and trivial.

>
>Look: There are less than 10^100 Chinese charaters. You may inform

did you count them ? What dictionary did you count them in ? Why did you
count them not in a linear order ?

which comes first in your count, chinese for water rat or chinese for
hedgehog ?

>Regards, WM

it is OK to fail, try again.
a good learning lesson for you.

[Virgil]

In article
<55f44998-1f0a-4d40...@y9g2000vbb.googlegroups.com>,

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

> On 11 Mrz., 11:32, fom <fomJ...@nyms.net> wrote:
>
>
> > If one then understands that "names" are (in principle)
> > introduced into mathematics by description, then those
> > names are introduced sequentially relative to the
> > Aristotelian sense of "that which is prior and that
> > which is posterior" inside of a deduction.
>
> Of course every name has to be explaned by other names or by showing
> the objects. In principle all senses including nose and ears are
> capable of perceiving definitions. It is an unhealthy state, induced
> by people who need cruches for thinking, that there should be only one
> written language called ZFC + FOPL.

Anyone who thinks that mathematics is limited to ZFC and or FOPL is the
one in need of crutches for thinking and everything else.

>
> Every idea that can be transferred and every carrier of this idea is
> capable of belonging to science.

Many of the ideas of religions, either theistic or WM-istic, are far to
anti-science to ever belong to it,


....

>
> If you want to form words you are in need of letters. If you cannot
> find them, you cannot use them. An alphabet without order is useless.

Chinese and Japanese are perfectly good languages without letters.

And all major languages were long spoken before being written, and thus
came into the world without letters.
--

[Virgil]

In article
<945a4b3b-5e37-4a7c...@r9g2000vbh.googlegroups.com>,

According to similarity transformations in plane geometry, the "number
of points" in a cube is the same as in a cube of twice the volume.

So, according to WM, similarity transformations must be in contradiction
with mathematics

[AMeiwes]

"WM" <muec...@rz.fh-augsburg.de> wrote in message

news:4451a7e8-9b2b-4657...@u2g2000vbx.googlegroups.com...

> On 11 Mrz., 09:17, fom <fomJ...@nyms.net> wrote:
>
>>
>> You have previously stated that in your reality
>> you cannot distinguish between the idea of
>> a number and the idea of a name.
>
> That is not correctly quoted.
> Numbers are names, but not all names are numbers.

numbers are simply dots on a page.

which type of numbering are you using?

all names are transmitted as a sequence of numbers, ASCI2

>>
>> > Or:
>> > Numbers without names cannot be well-ordered.
>> > All numbers can be well-ordered.
>>
>> You have previously denied the existence of
>> numbers without names.
>
> Of course, but here I talk about matheology which accepts and requires
> numbers without names. Did you really miss that?

since you cannot have names for numbers how can you tell someone each number

?

>
>> Your claims require you to use the axioms of set
>> theory (any commonly accepted version) and argue
>> according to methods that are generally recognized
>> to be representable in the deductive calculus of
>> first-order logic.
>
> Absolutely wrong. My claim is that a language is speakable, an
> alphabet is usable and a number is nameable. Any theory not obeying
> these facts is nonsense. There is no need to analyze its details.
> Wasted time.
>

Any language that is speakable, is mis-understandable, can be mis-spoken, or
not heard.
An alphabet is unusable to someone who does not know the alphebet.
since these two flaws are included in your statement about "Any theory not
obeying..." is also flawed.
the details prove your deductions using general statements are seriously
flawed.
yes, wasted time.



> Regards, WM

[Virgil]

In article
<e4e2a00f-9706-4036...@m4g2000vbo.googlegroups.com>,

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

> On 10 Mrz., 23:58, Virgil <vir...@ligriv.com> wrote:
>
> > 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)?
>
> ax is the binary sequence belonging to ax. What else should it be?
> If a is larger than 1, some binary sequences do not exist in the
> considered unit interval.

Then when the original linear space is the set of all binary sequences,
and x is in the space, ax need NOT be in that space?

That is a very odd sort of linear space by any standard,
or at least by any standard valid outside Wolkenmuekenheim.

>
> If f(x) is a path in the CIBT, then af(x) is the path belonging to
> af(x).

But how does one find which path af(x) is when given a and f(x)?

> >
> > Until these and a few other issues are settled,
>
> What is there to settle?

Showing that the set of binary sequences form a LINEAR space,
compatible with standard definitions of what constitutes a LINEAR space.

Showing that the set of paths of a CIBT form a LINEAR space,
compatible with standard definitions of what constitutes a LINEAR space.

Showing that some mapping from one to the other forms a LINEAR mapping,
compatible with standard definitions of what constitutes a LINEAR
mapping.

When WM has done these three things, only then will he be done.
So far WM has not done any of them.

If WM cannot do them all he should withdraw his claim that his mapping
is a linear mapping.

>
> Take, for the sake of simplicity, paths of the decimal tree and
> decimal sequences of the unit interval. Calculate exactly as you are
> used to do.
> If x = 0.1000... and a = 0.5000..., then ax = 0.05000...
> If f(x) = 0.1000... and a = 0.5000..., then af(x) = 0.05000...
> If desired convert into binaries.
>
> If you leave the unit interval, then the results are not belonging to
> the unit interval and not belonging to the tree.

If one even CAN leave the space that way, then one does not have a
linear space.

As any true mathematician would know.
--

[Virgil]

In article
<24048720-e6a3-4810...@x15g2000vbj.googlegroups.com>,

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

> On 10 Mrz., 23:42, "AMeiwes" <inva...@invalid.com> wrote:
> > "WM" <mueck...@rz.fh-augsburg.de> wrote in message
> >
> > news:5e27321f-996c-4ccc...@k14g2000vbv.googlegroups.com...
> > On 10 Mrz., 13:06, fom <fomJ...@nyms.net> wrote:> On 3/10/2013 5:07 AM, WM
> > wrote:
> >
> > <snipped>
> >
> > >Otherwise spare your comments.
> >
> > > Linear ordering does not imply countability.
> >
> > if the set has complex elements, how do you create linear ordering ?
>
> If the complex elements are discrete, one could use Cantor's zigzag
> method.

Consider the complex plane. Or points of an infinite dimensional space.

>
> But it is not at all the question how to create a linear ordering. An
> alphabet, by definition and by practical application, is not an
> alphabet unless it is discrete and linearly ordered, i.e., unless it
> is a sequence and, therefore, countable.
>
> Regards, WM

WM has 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
f(ax + by) = af(x) + bf(y), where a and b are arbitrary members of the
field of scalars and x and y are binary sequences and f(x) and f(y) are
paths in a CIBT.

By the way, WM, what are ax and 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)?

[Virgil]

In article
<6d86da85-aeae-4db5...@m4g2000vbo.googlegroups.com>,

[Virgil]

In article
<5771bf4f-1f15-4c96...@14g2000vbr.googlegroups.com>,

I do not believe that there is any standard order for the set of ALL
Chinese characters or the set of ALL Japanese characters, and even if
there were, it would be virtually impossible to memorize it.

>
> Look: There are less than 10^100 Chinese charaters. You may inform
> yourself here:
> http://en.wikipedia.org/wiki/Chinese_characters
>
> Regards, WM

[WM]

On 11 Mrz., 19:45, "AMeiwes" <inva...@invalid.com> wrote:

> >Look: There are less than 10^100 Chinese charaters.
>

> did you count them ?

No, I applied rational thinking and knowledge about our planet for an
upper estimation. Try to apply it too and make probable that you have
succeeded. Until then EOD.

Regards, WM

[AMeiwes]

"WM" <muec...@rz.fh-augsburg.de> wrote in message

news:7245b813-0ecb-4096...@r8g2000vbj.googlegroups.com...

Why did you require "planet knowledge" to solve this problem? This is a
cozmic problem ?


>
> Regards, WM

[AMeiwes]

"WM" <muec...@rz.fh-augsburg.de> wrote in message

news:80bc4329-3c7b-4477...@r8g2000vbj.googlegroups.com...

the Null Volume.

>Regards, WM

[AMeiwes]

"WM" <muec...@rz.fh-augsburg.de> wrote in message

news:6d86da85-aeae-4db5...@m4g2000vbo.googlegroups.com...

> On 10 Mrz., 23:50, Virgil <vir...@ligriv.com> wrote:
>
>> Lexical ordering of words, unless there is a fixed limit on world
>> length, will allow infinitely many words having no immediate
>> predecessors. Consider the problem of lexically ordering all finite
>> binary sequences with no limit on their finite lengths.
>>
>
> In an alphabet, however, every symbol except the first one, will have
> a predecessor, and every symbol, except the last one, will have a
> successor. Otherwise it cannot be applied and is not an applicable
> alphabet.

there is no need to have a first or last, it was just taught to you that way
when you were a child.

>
> Perhaps I should have said: I talk about usable alphabets for
> speakable languages (in case a matheologian might introduce his
> unusable alphabets for his unspeakable languages).

therefore a non-matheologian might not introduce his usable alphabets for
his non-unspeakable languages.

>
>> > How should an alphabet be used if it is not ordered and the symbols
>> > cannot be found?
>>
>> Consider written Chinese or Japanese, which have no alphabet of the sort
>> you demand.
>
> It is impossible to find a certain symbol other than searching all
> symbols?

depends upon the symbol.

> Perhaps that's why Chines is so difficult to learn. I don't know. But
> anyhow it is possible to put all symbols in a given sequence if there
> is at least one written catalogue of all symbols.

the real question is how many sequences are there ? far more than one.

>
> Regards, WM

[Virgil]

In article
<49740516-cc2b-4128...@w3g2000vba.googlegroups.com>,

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

> On 10 Mrz., 23:32, Virgil <vir...@ligriv.com> wrote:
>
> >
> > > > Linear ordering does not imply countability.
> >
> > > Linear ordering of an alphabet does.
> >
> > Only if one requires a finite alphabet.
>
> No.

Yes!

>
> > Since some of the most common linear orders in real mathematics are NOT
> > finite, much less discrete,
>
> Any usable alphabet is discrete. Unusable alphabets are not part of
> mathematics

But precision of definition is a part of mathematics. WM claimed that
linear ordering implies well ordering, which it does not, at least not
outside Wolkenmuekenheim .

If WM means it only for finite sets, for which it does hold, he must
specify finiteness expicitly, since outside of Wolkenmuekenheim,
finiteness is neither required or assumed in defining linear ordering of
sets, and the vast majority of the world lives totally outside
Wolkenmuekenheim.





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

[Virgil]

In article
<4451a7e8-9b2b-4657...@u2g2000vbx.googlegroups.com>,

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

> On 11 Mrz., 09:17, fom <fomJ...@nyms.net> wrote:
>
> >
> > You have previously stated that in your reality
> > you cannot distinguish between the idea of
> > a number and the idea of a name.
>
> That is not correctly quoted.
> Numbers are names, but not all names are numbers.

WRONG! Number HAVE names, but a number is no more ONLY a name than a
person is only a name.

This is proved by the fact that one number may have many different names.

> >
> > > Or:
> > > Numbers without names cannot be well-ordered.
> > > All numbers can be well-ordered.
> >
> > You have previously denied the existence of
> > numbers without names.
>
> Of course, but here I talk about matheology which accepts and requires
> numbers without names. Did you really miss that?

Since numbers are not mere names, even though WM is too inane to be
aware of it, why should it be surprising that there are numbers without
any names?

It should not, at least outside of the miasma we call Wolkenmuekenheim.

>
> > Your claims require you to use the axioms of set
> > theory (any commonly accepted version) and argue
> > according to methods that are generally recognized
> > to be representable in the deductive calculus of
> > first-order logic.
>
> Absolutely wrong. My claim is that a language is speakable, an
> alphabet is usable and a number is nameable. Any theory not obeying
> these facts is nonsense. There is no need to analyze its details.
> Wasted time.

Not all languages are "speakable". The sign languages of the deaf are in
a very real sense unspeakable and some computer languages, especially
the machine languages, are, in a quite different way, unspeakable.

Thus it may also be that some alphabets are unusable, and it is
certainly the case that everywhere but in Wolkenmuekenheim there are
numbers which have no names, if for no other reason that everywhere but
Wolkenmuekenheim there are more numbers than names.
>
> Regards, WM






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

WM has 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
f(ax + by) = af(x) + bf(y), where a and b are arbitrary members of the
field of scalars and x and y are binary sequences and f(x) and f(y) are
paths in a CIBT.

By the way, WM, what are ax and 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.

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

[Virgil]

In article
<80bc4329-3c7b-4477...@r8g2000vbj.googlegroups.com>,

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

> On 10 Mrz., 21:03, William Hughes <wpihug...@gmail.com> wrote:
> > On Mar 10, 1:33�pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > On 10 Mrz., 12:55, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > In Wolkenmuekenheim a procedure that is point preserving
> > > > is volume preserving.
> >
> > Question: �Let f be a bijection. �Is f point preserving?
>
> Question: Can there exist a volume without any point in it?
>

> Regards, WM

In Wolkenmuekenheim there exist volumes without any point to them.

[WM]

On 11 Mrz., 20:13, Virgil <vir...@ligriv.com> wrote:

>
> Then when the original linear space is the set of all binary sequences,
> and x is in the space, ax need NOT be in that space?

The original domain is only a part of a linear space. Also for a part
I can define isomorphism, as I have done. Of course I could extend
tree and interval to the complete real space. But I do not wish to do
so.

Regards, WM

[WM]

On 11 Mrz., 20:26, Virgil <vir...@ligriv.com> wrote:

>
> I do not believe that there is any  standard order for the set of ALL
> Chinese characters or the set of ALL Japanese characters, and even if
> there were, it would be virtually impossible to memorize it.

But it is possible to write them in a dictionary, as has been done,
and to find them, by some technique unknown to me.
http://en.wikipedia.org/wiki/Chinese_characters#Number_of_Chinese_characters

Regards, WM

[fom]

On 3/11/2013 7:36 AM, WM wrote:
> On 11 Mrz., 11:32, fom <fomJ...@nyms.net> wrote:
>
>
>> If one then understands that "names" are (in principle)
>> introduced into mathematics by description, then those
>> names are introduced sequentially relative to the
>> Aristotelian sense of "that which is prior and that
>> which is posterior" inside of a deduction.
>
> Of course every name has to be explaned by other names or by showing
> the objects.

One can formulate the basic relations of set theory using
self-referencing syntax. The only sense of "showing" is
what can be formulated by sentences admissible to the
grammar of the deductive calculus.

AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))

AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))

AxAy((Az(ycz -> xez) /\ Ez(xez /\ -ycz)) -> Az((xez /\ -ycz) -> (Ew(xew
/\ wcy) \/ Aw(zcw -> ycw))))

What you describe is the kind of non-sensical syntactic
priority that reverts to the use of "undefined language primitives"

With regard to that, both Aristotle and Frege made the same
mistake. Aristotle spoke of "essence" and conflated it with
"substance". Frege spoke of "truth" and conflated it with
"existence".

The Aristotelian distinction between demonstrative science
and dialectical argumentation is sharp except for the involvement
of those principles.

Since the modern paradigm has distinguished between mathematics
and empirical sciences, Aristotle's sharp distinction can be
employed usefully. The distinction, however, is based upon
an epistemic justification. And, the modern metamathematics
is firmly entrenched in what is no less than an irrational
hatred of epistemology.

In the historical period, it was the search for "simple" substance
to ground mathematics that led to "undefined language primitives."
Leibniz applied Aristotelian logic appropriate to Aristotle's
description of how a necessary principle is structured. By the
time of Bolzano, the limited form of the syllogistic logic and
adherence to a principle of syntactic priority in definition
introduced the problem of undefined language primitives.

With regard to the modern period, it was the problem of existence
in the guise of presuppostion failure. That was the basis of
Russell's rejection of Fregean description theory. Although
Russell's description theory -- treating descriptions as a form
of quantifier -- resolved the problem of presupposition failure,
it did so at the cost of the traditional epistemic qualities
that characterize "definition".

If one restricts "formal" mathematics to what can occur among
sentences admissible to the deductive calculus, then "essence"
is no more than the definiens of the "first" description in
the sequential order of the derivation. Whether or not two
names correspond to a single purported referent retains its
traditional epistemic character. Certain other restrictions
are necessary. However, the point is that the nature of
mathematics in which the facts of the science relate through
deductive reasoning is given priority over whether anyone
materially believes those facts.

I assure you that if God is responsible for the natural numbers,
that gift was not given to be used as a club (Hobbes: the only
use for mathematics is war). Mathematical form ought not be
the basis for material physicalism, and, that is certainly
one interpretation of Frege's assertion that logic is about
truth.

> In principle all senses including nose and ears are
> capable of perceiving definitions.

Sure. But, with regard to objective forms, one must reject that
as a foundational ground. In Carnap, one clearly sees the
influence Piercean semiotics whereby everything is a sign and
the system of signs takes an infinite regress into non-grammatical
forms. The alternative is Saussere. For Saussere, there is an
"observable sign vehicle." In the case of mathematics -- because
of the abstract nature of its objects -- that "observable sign
vehicle" needs to be a piece of grammatical syntax.

> It is an unhealthy state, induced
> by people who need cruches for thinking, that there should be only one
> written language called ZFC + FOPL.

I would not put it in quite that manner. It is unfortunate,
however, that this material is being taught as if the original
sources can be ignored.

Since my views concerning identity have led to non-standard
interpretation of both of your references, I cannot refer to
those references as others might.

Aristotle is clear that his logic does not treat of the "parts"
of its individuals. There is a very real sense by which the
advances of the nineteenth century began to correct this problem.
But, among other things, it introduces essential infinities.

The relationship between Leibniz' original description of the
identity of indiscernibles and the definite article so critical
to a description theory is Cantor's intersection theorem. Leibniz
clearly refers to geometers and "specific differences" in their
figures. Leibniz' original remarks have been effectively censored
by the historical interpretations of "arithmetization" and
"logicism".

The only way to relate "parts" to their "wholes" in this sense
is through the topological aspects of Cantor's research. And,
in relation to description theory, individuation is properly
characterized by a nested sequence of closed non-empty sets
with vanishing diameters. To be a defined object in the sense
of a definite description making singular reference is to
cross Zeno's finishing line.

As for the logical side of things, Frege's system treats of
individuals through direct grammatical reference. It does
not treat individuals in relation to the grammatical structure
of a sentence as the Aristotelian system does. So, quite
naturally, the interpretation of the quantifiers becomes
subject to revision. While it has been some time, I seem
to recall that Frege originally interpreted the universal
quantifier according to a "choice" interpretation rather
than a "course of values" interpretation. This resulted
in an arbitrary designation of falsity for nonsensical
references. Once again, the influence of Russell's objections
and Russell's description theory come into play.

These nineteenth century advances ought to be respected.
What seems lost with the current situation in metamathematics
is that logical research did not stop with Russell. Strawson
was the first substantive critic of Russellian description
theory, and, his characterization of numerical identity
reintroduces Leibniz' original geometric sensibility.
Robinson also criticized Russellian description theory, and,
his analysis reintroduced the sense that names construct
the diagonal of models and, thereby, ground the completion
of Fregean incomplete symbols. On a more modern note, the
study of descriptions is finally addressing the notion of
a descriptively-defined name. Substantively, the classic
example is the planet of Uranus to which reference had been
established prior to its material existence. But, the
analysis of names introduced solely by description clearly
applies to mathematical contexts.

It was Carnap, a follower of Russell, who was forced to
introduce the distinction between pragmatics and semantics
in order to protect his own "ideal language theory" from
legitimate criticisms as to applicability. So, it is
intellectually humorous for me to watch a metamathematics
based on "undefined language symbols" interpreted as
"definitions in use" assert that the meaning of their
symbols is given by "semantics".

This is why I pursued an understanding of mathematics
in which every relation, function and constant is
given by a definition (relative to the form of the
deductive calculus). In this regard I am far more
the subject of your criticism than even those who promote
ZFC and FOPL.

Now, to really mix things up in the scenario described
above, throw in the objections and mathematics of the
finitists such as yourself.

>
> Every idea that can be transferred and every carrier of this idea is
> capable of belonging to science. There is not the least reason to
> believe that a formal proof in written language is preferable to
> spoken words or to drawings or pictures.

It is not. In other posts, I have had people confused about
certain statements that I make. Each time, I assert the fact
that mathematics ought to be practiced precisely as it is
practiced in classrooms and mathematician's offices. What
you assert here is that mathematicians are human beings who
communicate according to the *pragmatics* of language.

To understand how the logicians helped mathematicians, one need
only look to a subject called "graphic statics" in which structural
specifications had been estimated from carefully drawn force diagrams.

One may surmise that a certain amount of geometry had been done
according to such methods. Prior to the elevation of standards
brought about by "arithmetization" and calls for logical rigor
in the nineteenth century, such methods seemed to have
occasionally asserted false claims in geometry.

Even as recently as the 1980's, there are instances. I had
a real analysis professor ask for a proof concerning a mapping
from the Cantor square into the interval [-1,1]. What he graded
as "excellent" were simply drawings showing that a 45 degree
line intersecting the initial configuration also intersected its
first iteration. My proof, of course, was not graded "excellent".
It had been written analytically and proved that any given
45 degree line intersecting the initial configuration also
intersected the Cantor dust of the limit. You might, quite
naturally, agree with my former professor.

For my part, I tire of metamathematical positions based on
logicism from people who "represent" mathematics, but do
not consider what practice with respect to those representations
actually dictates.

To get a sense of what I mean, here is one very ugly construction
of Dedekind cuts within set theory,

news://news.giganews.com:119/M5qdncGgG5a-VbvM...@giganews.com

Once again, long and unwieldly. So do not waste too much time.

But, the lesson to be learned is that there is no sense of a
"name", a "description", or even a "choice function" if one
actually tries to do the work within the constraints of the
formalisms. As George Greene once observed, there is no
model construction language.

>>
>> Well, if you have bothered to read this
>> far,
>
> In fact, I did - with interest.
>

I am glad for that. I dislike the internet because I find myself
engaging in ad hominem attacks that are, at best, unproductive and,
at worst, demonstrate an embarassing foolishness.

In spite of our divergent views on the nature of mathematics,
let me take a moment to apologize for those that have been
directed toward you.

[WM]

On 11 Mrz., 22:38, Virgil <vir...@ligriv.com> wrote:
> In article
> <4451a7e8-9b2b-4657-91b7-1880bdb93...@u2g2000vbx.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 11 Mrz., 09:17, fom <fomJ...@nyms.net> wrote:
>
> > > You have previously stated that in your reality
> > > you cannot distinguish between the idea of
> > > a number and the idea of a name.
>
> > That is not correctly quoted.
> > Numbers are names, but not all names are numbers.
>
> WRONG! Number HAVE names, but a number is no more ONLY a name than a
> person is only a name.
>
> This is proved by the fact that one number may have many different names.

That is a philosophical question that you may decide in your sense.
One could say: A number is the union of all its names and possibly
further properties abd rules determining it. But in matheology, there
are numbers that neither are nor have names.

Regards, WM

[Virgil]

In article
<7245b813-0ecb-4096...@r8g2000vbj.googlegroups.com>,

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

> On 11 Mrz., 19:45, "AMeiwes" <inva...@invalid.com> wrote:
>
> > >Look: There are less than 10^100 Chinese charaters.
> >
> > did you count them ?
>
> No, I applied rational thinking and knowledge about our planet for an
> upper estimation.

WM is so rarely rational hat one forgets that on rare occasions he can
be.

But only when the issue of finiteness versus non-finiteness has not come
up.

[fom]

On 3/11/2013 1:46 PM, Virgil wrote:
> In article
> <55f44998-1f0a-4d40...@y9g2000vbb.googlegroups.com>,
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>> On 11 Mrz., 11:32, fom <fomJ...@nyms.net> wrote:
>>
>>
>>> If one then understands that "names" are (in principle)
>>> introduced into mathematics by description, then those
>>> names are introduced sequentially relative to the
>>> Aristotelian sense of "that which is prior and that
>>> which is posterior" inside of a deduction.
>>
>> Of course every name has to be explaned by other names or by showing
>> the objects. In principle all senses including nose and ears are
>> capable of perceiving definitions. It is an unhealthy state, induced
>> by people who need cruches for thinking, that there should be only one
>> written language called ZFC + FOPL.
>
> Anyone who thinks that mathematics is limited to ZFC and or FOPL is the
> one in need of crutches for thinking and everything else.

But, he is correct about this Virgil.

10 years ago I came to these newsgroups without knowledge
of what constitutes "mathematics" in certain academic programs.
With my first post I was targeted for a flame. Try as I might
to engage any considerations that might lie outside of the
dogma of FOPL, it did not happen. Although I tried to be
civil, I had been mercilessly flamed for two years until finally
the posts started to allege racism. Then I just left.

My current state of philosophical knowledge is the response
to that flame (at the expense of what little mathematical
skill that remains -- a high price).

What he is saying is within certain traditions.

Leibniz called for a "philosophical calculus" that could decide
truth.

Peano called for a unified language so that difficulties of
colloquial languages could be minimized.

Principia Mathematica is precisely an attempt along these
lines.

Even where those agendas are not explicit, the real problem is
that ZFC + FOPL becomes a simplistic fallback position for
dismissing non-standard opinions. Associated foundational
claims make them "inviolate" in such circumstances.


It is certainly refreshing to see your statement.

[Virgil]

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

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

> On 3/11/2013 1:46 PM, Virgil wrote:
> > In article
> > <55f44998-1f0a-4d40...@y9g2000vbb.googlegroups.com>,
> > WM <muec...@rz.fh-augsburg.de> wrote:
> >
> >> On 11 Mrz., 11:32, fom <fomJ...@nyms.net> wrote:
> >>
> >>
> >>> If one then understands that "names" are (in principle)
> >>> introduced into mathematics by description, then those
> >>> names are introduced sequentially relative to the
> >>> Aristotelian sense of "that which is prior and that
> >>> which is posterior" inside of a deduction.
> >>
> >> Of course every name has to be explaned by other names or by showing
> >> the objects. In principle all senses including nose and ears are
> >> capable of perceiving definitions. It is an unhealthy state, induced
> >> by people who need cruches for thinking, that there should be only one
> >> written language called ZFC + FOPL.
> >
> > Anyone who thinks that mathematics is limited to ZFC and or FOPL is the
> > one in need of crutches for thinking and everything else.
>
> But, he is correct about this Virgil.

Even if WM did manage to be correct about something, he would almost
certainly do so for the wrong reasons, as any prolonged reading of his
posts will verify.

>
> 10 years ago I came to these newsgroups without knowledge
> of what constitutes "mathematics" in certain academic programs.
> With my first post I was targeted for a flame. Try as I might
> to engage any considerations that might lie outside of the
> dogma of FOPL, it did not happen. Although I tried to be
> civil, I had been mercilessly flamed for two years until finally
> the posts started to allege racism. Then I just left.
>
> My current state of philosophical knowledge is the response
> to that flame (at the expense of what little mathematical
> skill that remains -- a high price).
>
> What he is saying is within certain traditions.
>
> Leibniz called for a "philosophical calculus" that could decide
> truth.
>
> Peano called for a unified language so that difficulties of
> colloquial languages could be minimized.
>
> Principia Mathematica is precisely an attempt along these
> lines.
>
> Even where those agendas are not explicit, the real problem is
> that ZFC + FOPL becomes a simplistic fallback position for
> dismissing non-standard opinions. Associated foundational
> claims make them "inviolate" in such circumstances.
>
>
> It is certainly refreshing to see your statement.

--

[Virgil]

In article
<68492561-fda3-413d...@c10g2000vbt.googlegroups.com>,

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

> On 11 Mrz., 22:38, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <4451a7e8-9b2b-4657-91b7-1880bdb93...@u2g2000vbx.googlegroups.com>,
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 11 Mrz., 09:17, fom <fomJ...@nyms.net> wrote:
> >
> > > > You have previously stated that in your reality
> > > > you cannot distinguish between the idea of
> > > > a number and the idea of a name.
> >
> > > That is not correctly quoted.
> > > Numbers are names, but not all names are numbers.
> >

> > WRONG! Numbers HAVE names, but a number is no more ONLY a name than a

> > person is only a name.
> >
> > This is proved by the fact that one number may have many different names.
>
> That is a philosophical question that you may decide in your sense.

It has long since been decided to be true.

NO name, at least in mathematics rather than WMytheology, is the thing
it names.

> One could say: A number is the union of all its names and possibly
> further properties abd rules determining it.

One can say any number of things, even mutually contradictory things,
as WM so often does, but saying something does not make it true,
even though WM seems to think he can make things true by merely saying
them over and over ad nauseam.

> But in matheology, there
> are numbers that neither are nor have names.

That is the case everywhere outside of WMytheology, not only in WM's
alleged and mythical land of matheology.
--

[Virgil]

In article
<3ef98ab9-4407-4d79...@g8g2000vbf.googlegroups.com>,

That same WIKI article states that none of the dictionaries listed can
be regarded as complete because the nature of the written Chinese
language is such that new characters are continuously being created.

One may also note that there is no coherent rule for connecting any
written characters with their spoken equivalents. At one time, there
were many mutually incomprehensible dialects all using different sounds
for the same written word/character. Lately that is fading out somewhat
with the enforcement of the Mandarin dialect as the official spoken
language.







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

[Virgil]

In article
<1e0619cf-7346-4137...@y9g2000vbb.googlegroups.com>,

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

> On 11 Mrz., 20:13, Virgil <vir...@ligriv.com> wrote:
>
> >
> > Then when the original linear space is the set of all binary sequences,
> > and x is in the space, ax need NOT be in that space?
>
> The original domain is only a part of a linear space.

WM has not yet defined the linear space structure of which the set of
all binary sequences is alleged to be a subset (but apparently not a
subspace).

WM has not yet defined the linear space structure of which the set of
paths of a CIBT is alleged to be a subset (but apparently not a
subspace).

WM has not yet described the mapping between those
subset-but-not-subspace structures, nor shown it to be linear either on
those subset-but-not-subpace parts or on any whole spaces of which they
are subsets.

> Also for a part
> I can define isomorphism, as I have done.

What sort of isomorphism? Unless it is an isomorphism of at least the
linear space structures, it is irrelevant to demonstrating the
linearity of the mapping under discussion.

For someone claiming to be a mathematician to be so obviously ignorant
of even the basics of linear spaces and their linear mappings
(homomorphisms) is both startling and depressing!

> Of course I could extend
> tree and interval to the complete real space. But I do not wish to do
> so.

Until WM has made both the set of binary sequences and the set of paths
of a CIBT into at least subsets, even if not entire subspaces of linear
spaces and shown that a map between them is linear at least on those
subsets, he has claimed what it now appears he cannot prove.

But for WM to prove what he origianally claimed, he should show both set
to be linear spaces and a mapping which is actually a linear mapping
between two entire spaces.


It is nowhere nearly the first time that WM has made a claim which he
has not been able to prove.

But, as so often, WM has made a claim that he seems too incompetent to
support and too arrogant to withdraw.
--

[fom]

On 3/11/2013 5:32 PM, Virgil wrote:
> In article
> <7245b813-0ecb-4096...@r8g2000vbj.googlegroups.com>,
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>> On 11 Mrz., 19:45, "AMeiwes" <inva...@invalid.com> wrote:
>>
>>>> Look: There are less than 10^100 Chinese charaters.
>>>
>>> did you count them ?
>>
>> No, I applied rational thinking and knowledge about our planet for an
>> upper estimation.
>
>
> WM is so rarely rational hat one forgets that on rare occasions he can
> be.
>
> But only when the issue of finiteness versus non-finiteness has not come
> up.
>

I took the time to excerpt some items
from Markov. You and William have made
lesser or greater efforts to understand
WM's notions concerning quantification
and induction. The following excerpts may
provide some insight on how he implements
his finitism relative to someone who has
made the effort to systematize these
notions.

One thing that should be noted when
considering the following excerpts
is that Frege originally interpreted
universal quantification in terms of
arbitrary selections. Markov's
explanations seem to be returning to
that interpretation by virtue of his
use of "given". Historically, this
is one aspect of Frege's work that
had been objected to by Russell. Thus
in so far as "classical" mathematics
follows Russell's program, it treats
universal quantification along the lines
of a course of values rather than as an
arbitrary selection.

A second thing to remember is that the
construction of syntactic forms from finite
alphabets using concatenations serves as
a ground for objectively accepted knowledge
as it pertains to the kind of objects
that may be generally given in his
formulation of a constructive mathematics.



From "Theory of Algorithms" by Markov:


"Abstraction of Potential Feasibility

"1. Carrying out constructive processes,
we often come up against obstacles
connected with a lack of time, space,
and material. [...]

"Acting in this manner, we shall disregard
the restrictedness of our possibilities
concerning time, space, and material.
This disregarding is customarily called
the abstraction of potential feasibility.
[...]

"4. Every application of potential
feasibility is an imaginative act. This
is equally true for any of the abstractions
inherent in mathematics or some other
abstract science. Classical mathematics
invokes abstractions going much farther
than those of constructive mathematics. In
particular, it makes use of the abstraction
of actual infinity, i.e., allows itself to
reason about 'infinite sets' as about given
non-constructive 'objects'. The difference
between 'classicists' and 'constructivists'
consists in their being willing to
accept abstractions of different sorts.
[...]



"Universal Statements

"1. An important role in mathematics is
played by universal statements beginning
with the words <<every>>, <<for every>>,
<<whatever be>>,etc. How can they be
understood constructively?

"2. The problem is simply solved when there
is a list of all objects of the kind to
which the universal statement refers. In
this case, the statement that every object
of this kind satisfies a given requirement
can be understood as a many-term
conjunction, each of whose terms asserts
this about one object in the list, where
all objects in the list appear in the
conjunction in the sense indicated.

"[...] But if the list contains only one
name, we shall then, of course, understand
the universal statement as the statement,

<<the unique object named in the list
satisfies the given requirement>>

"[...] in the case when the list is
empty, i.e., when there are no objects
of this kind. We shall then understand
our statement as the trivial truth,

nullstring *=* nullstring

[*=* is "graphic equality"]

(independently of what the requirement
which an object should satisfy consists
of).

"With such an understanding of universal
statements in those cases when there is
such a list of objects of the kind under
consideration, the following holds: When
adding a single new object to this kind,
the statement that every object of the
extended kind satisfies the formulated
requirement turns out equivalent to the
conjunction of the previous universal
statement (pertaining to the initial
kind) and the statement that the added
object satisfies this requirement.

"3. Everything said in part 2 pertained to
the case when there is a list of objects
of the kind under consideration. There
may, however, be no such list, or it
may even be unfeasible because there are
'too many' of these objects.

"Fortunately, there is another possible
way of understanding universal statements
constructively. For certain objects of
the kind under consideration, it may be
established by means of suitable arguments
that they satisfy the formulated
requirments. These arguments may be of
such a nature that the possibility of
developing analogous arguments for any
other object of the kind under consideration
will be clear. A conviction will then
grow up in us that whatever object of this
kind given to us, we shall be able to
prove (make obvious through arguments) that
this object satisfies the requirement
presented. We are claiming that precisely
this situation prevails when we say that
every object of the given kind satisfies
the given requirement.

"Thus, we have arrived at the following
understanding of statements to the effect
that every constructive object of a given
kind satisfies a given requirement. Such
a statement claims our ability to prove
for any given object of the kind under
consideration, that it satisfies the
requirement presented.

"The source of our confidence in being
able to carry out the required proof,
regardless of which object we are given,
may be our limited experience with such
proofs, provided that this experience has
made it clear how we should act in an
arbitrary case. This ability of ours to
become convinced of the truth of universal
statements on the basis of limited
experience is what we call our intuition
of the universal. [...]



"Direct Negation. Decidable Statements

"1. Studying our ability to carry out
constructive processes, we formulate
the results of this study in the form of
certain statements asserting that at the
present time we are able to construct
certain objects, we have mastered certain
general methods, etc. It is natural
to ask what the negations of statements
of this kind can look like, negations
also saying something about our constructive
abilities.

"A naive answer says that the negation
of the statement (1)

<<at the present time we are able to
construct an object satisfying the
requirement...>>

is the statement (2)

<<at the present time we are unable to
construct an object satisfying the
requirement...>>

that the negation of the statement (3)

<<at the present time we possess a
general method for...>>

is the statement (4)

<<at the present time we do not possess
a general method for...>>


"This answer must, however, be rejected
because statements of the form (2) and
(4) may well turn out to be true today
and false tomorrow. [...]

"We see, therefore, that a naive
treatment of negation can take us
beyond the bounds of [constructive]
mathematics. Desiring to stay within
these bounds, we should be concerned
about acquiring a suitable positive
understanding of negation. [...] In
this connection, we shall naturally
require that the negation of a statement
always be incompatible with it, i.e.
that the conjunction of a statement
with its negation be false. [...]

"In general, in the case when statements
A and B are such that their conjunction
is false, but their disjunction is
true, we shall say that B is the
direct negation of A. We shall say that
a statement A is decidable when we
have succeeded in selecting its direct
negation.

"In case a statement A is decidable,
we have a method for determining whether
A is true. We determine this by
establishing the truth of the disjunction

<<A or B>>

where B is the direct negation of A.
[...]


"Semidecidable Statements. Strengthened Negation

"1. Let us now consider a statement about
the existence of a word in a given alphabet
satisfying a given requirement expressed
by a decidable predicate, i.e., a
statement (1)

<<there exists an X such that F>>

Where X is a free verbal variable and F is
a decidable, one-place predicate with this
variable.

"We shall call statements of this kind
semidecidable. [...]

"Investigations analogous to those just
carried out can obviously be performed
for any semidecidable statement. They
suggest that we regard the statement (5)

<<given any X, G holds>>

where G is a direct negation of F, to
be a direct negation of the semidecidable
statement (1).

"The statement (5) will be called a
strengthened negation of the statement (1).
[...]



"Material Implication

"[...] Thus we call the implication

<<if A, then B>>

understood as the disjunction

<<C or B>>

where C is a direct negation of A, the
material implication with premise A and
conclusion B. [...]



"Strengthened Implication

"1. Let us now consider implications with
a semidecidable premise, i.e., statements
of the form (1)

<<if there exists an X such that F,
then A>>


"[...] Let us try to interpret the implication
(1) in the spirit of material implications
as the disjunction (2)

<<at the present time we do not possess a
method for constructing a word in the given
alphabet, satisfying the predicate F, or A>>


"One finds that this disjunction can be true
at a given moment, but is not insured
against refutation in the future. This
will be the case when A is false and we are
unable to construct a desired word now,
but shall be able to do so later. [...]

"Note, however, that the statement (2)
would be insured against refutation if
the statement (3)

<<given any X, if F, then A>>

were proven, where the implication obtained
from the phrase (4)

<<if F, then A>>

after replacement of the variable X [in F]
by any word in our alphabet should be
understood as a material implication. [...]

"All this suggests that precisely the
statement (3) be considered as the
interpretation of the implication (1).
We shall call the implication (1),
understood in this way, the strengthened
implication with the premise

<<there exists an X such that F>>

and the conclusion A.

"According to the definition, a strengthened
implication always has a semidecidable
premise."

[fom]

On 3/11/2013 7:12 PM, Virgil wrote:
> In article <nPadnaCGpdf_waPM...@giganews.com>,
> fom <fom...@nyms.net> wrote:
>
>> On 3/11/2013 1:46 PM, Virgil wrote:
>>> In article
>>> <55f44998-1f0a-4d40...@y9g2000vbb.googlegroups.com>,
>>> WM <muec...@rz.fh-augsburg.de> wrote:
>>>
>>>> On 11 Mrz., 11:32, fom <fomJ...@nyms.net> wrote:
>>>>
>>>>
>>>>> If one then understands that "names" are (in principle)
>>>>> introduced into mathematics by description, then those
>>>>> names are introduced sequentially relative to the
>>>>> Aristotelian sense of "that which is prior and that
>>>>> which is posterior" inside of a deduction.
>>>>
>>>> Of course every name has to be explaned by other names or by showing
>>>> the objects. In principle all senses including nose and ears are
>>>> capable of perceiving definitions. It is an unhealthy state, induced
>>>> by people who need cruches for thinking, that there should be only one
>>>> written language called ZFC + FOPL.
>>>
>>> Anyone who thinks that mathematics is limited to ZFC and or FOPL is the
>>> one in need of crutches for thinking and everything else.
>>
>> But, he is correct about this Virgil.
>
> Even if WM did manage to be correct about something, he would almost
> certainly do so for the wrong reasons, as any prolonged reading of his
> posts will verify.

Right and wrong are probably subjective. But, I
understand you perfectly well.

[WM]

On 11 Mrz., 22:58, fom <fomJ...@nyms.net> wrote:
>
> In spite of our divergent views on the nature of mathematics,
> let me take a moment to apologize for those that have been
> directed toward you.

Let me reciprocate that apology.

Regards, WM

[WM]

On 12 Mrz., 01:24, Virgil <vir...@ligriv.com> wrote:

> > But in matheology, there
> > are numbers that neither are nor have names.
>
> That is the case

And how do you put them into any order?

Regards, WM

[WM]

On 12 Mrz., 01:38, Virgil <vir...@ligriv.com> wrote:
> In article

> <3ef98ab9-4407-4d79-b01c-1cb49a40d...@g8g2000vbf.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 11 Mrz., 20:26, Virgil <vir...@ligriv.com> wrote:
>
> > > I do not believe that there is any  standard order for the set of ALL
> > > Chinese characters or the set of ALL Japanese characters, and even if
> > > there were, it would be virtually impossible to memorize it.
>
> > But it is possible to write them in a dictionary, as has been done,
> > and to find them, by some technique unknown to me.

> >http://en.wikipedia.org/wiki/Chinese_characters#Number_of_Chinese_cha...

>
>
>
> That same WIKI article states that none of the dictionaries listed can
> be regarded as complete because the nature of the written Chinese
> language is such that new characters are continuously being created.

A wonderful example for potential infinity!

>
> One may also note that there is no coherent rule for connecting any
> written characters with their spoken equivalents.

That is irrelevant with respect to their shape and the problem of
ordering them sequentially.

Regards, WM

[WM]

On 12 Mrz., 02:06, Virgil <vir...@ligriv.com> wrote:
> In article

> <1e0619cf-7346-4137-a6d0-128dd0116...@y9g2000vbb.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 11 Mrz., 20:13, Virgil <vir...@ligriv.com> wrote:
>
> > > Then when the original linear space is the set of all binary sequences,
> > > and x is in the space, ax need NOT be in that space?
>
> > The original domain is only a part of a linear space.
>
> WM has not yet defined the linear space structure of which the set of
> all binary sequences is alleged to be a subset (but apparently not a
> subspace).
>
> WM has not yet defined the  linear space structure of which the set of
> paths of a CIBT is alleged to be a subset (but apparently not a
> subspace).
>
> WM has not yet described the mapping between those
> subset-but-not-subspace structures, nor shown it to be linear either on
> those subset-but-not-subpace parts or on any whole spaces of which they
> are subsets.

Obviously the unit interval is a subset of |R. And the Binary Tree is
a subtree of the complete real tree that contains all real numbers
(extend the tree to the top).

> > Also for a part
> > I can define isomorphism, as I have done.
>
> What sort of isomorphism?

I had defined that. Look it up.

Regards, WM

[Virgil]

In article
<7447638f-0f19-4d79...@r9g2000vbh.googlegroups.com>,

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

> On 12 Mrz., 02:06, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <1e0619cf-7346-4137-a6d0-128dd0116...@y9g2000vbb.googlegroups.com>,
> >
> > �WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 11 Mrz., 20:13, Virgil <vir...@ligriv.com> wrote:
> >
> > > > Then when the original linear space is the set of all binary sequences,
> > > > and x is in the space, ax need NOT be in that space?
> >
> > > The original domain is only a part of a linear space.
> >
> > WM has not yet defined the linear space structure of which the set of
> > all binary sequences is alleged to be a subset (but apparently not a
> > subspace).
> >
> > WM has not yet defined the �linear space structure of which the set of
> > paths of a CIBT is alleged to be a subset (but apparently not a
> > subspace).
> >
> > WM has not yet described the mapping between those
> > subset-but-not-subspace structures, nor shown it to be linear either on
> > those subset-but-not-subpace parts or on any whole spaces of which they
> > are subsets.
>
> Obviously the unit interval is a subset of |R.

That does not make the set of all binary sequences into a subset of any
linear space until you explicitly specify or identify that space.

> And the Binary Tree is
> a subtree of the complete real tree that contains all real numbers
> (extend the tree to the top).

That does not make the set all paths of a CIBT into a subset of any
linear space until you explicitly specify or identify that space.

>
> > > Also for a part
> > > I can define isomorphism, as I have done.
> >
> > What sort of isomorphism?
>
> I had defined that. Look it up.

There are all sorts of isomorphisms, many, if not most, of which are
totally irrelevant to the problem at hand, and it does not appear that
WM has any idea which ones are relevant.

As far as we can tell, WM might be describing an isomorphism of
non-commutative groups, which are nothing like linear spaces or subsets
of linear spaces, and no isomorphism between non-commutative groups
could ever be a linear mapping.

It is clear that WM is trying, though vainly, to fake his way out of
trouble, and has no idea of what is really going on.
>
> Regards, WM

So WM, who tries to tell everyone how to run mathematics, does not seem
to have any understanding of linear spaces are or what linear mappings
are.

That someone so obviously ignorant of basic linear algebra should be
allowed to be a teacher of mathematics at anything above a grade school
level is unfair to his students.
--

[Virgil]

In article
<ae166029-cb2b-47fa...@h11g2000vbf.googlegroups.com>,

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

> On 12 Mrz., 01:38, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <3ef98ab9-4407-4d79-b01c-1cb49a40d...@g8g2000vbf.googlegroups.com>,
> >
> > �WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 11 Mrz., 20:26, Virgil <vir...@ligriv.com> wrote:
> >
> > > > I do not believe that there is any �standard order for the set of ALL
> > > > Chinese characters or the set of ALL Japanese characters, and even if
> > > > there were, it would be virtually impossible to memorize it.
> >
> > > But it is possible to write them in a dictionary, as has been done,
> > > and to find them, by some technique unknown to me.
> > >http://en.wikipedia.org/wiki/Chinese_characters#Number_of_Chinese_cha...
> >
> >
> >
> > That same WIKI article states that none of the dictionaries listed can
> > be regarded as complete because the nature of the written Chinese
> > language is such that new characters are continuously being created.
>
> A wonderful example for potential infinity!

Not even close!

It will always be measurably finite, since there is always a specific
and finite upper limit which can be places on the number of such words.
Which limit is under one trillion now and will remain so forever, as
humanity will go exist before that limit is threatened.

[Virgil]

In article
<aa3b230d-08a5-4e88...@h11g2000vbf.googlegroups.com>,

Nothing in any rules for the reals says that we have to be able to, it
is only required that such an order exists.

While for any two given different reals we know that one is necessarily
smaller that the other, nothing in the rules for the reals says that we
have to be able to tell which one is smaller. At least not without
knowing a good deal more about each number that just that it exists.

[WM]

On 12 Mrz., 10:05, Virgil <vir...@ligriv.com> wrote:
> In article

> <aa3b230d-08a5-4e88-85a0-8b5345590...@h11g2000vbf.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 12 Mrz., 01:24, Virgil <vir...@ligriv.com> wrote:
>
> > > > But in matheology, there
> > > > are numbers that neither are nor have names.
>
> > > That is the case
>
> > And how do you put them into any order?
>
> Nothing in any rules for the reals says that we have to be able to, it
> is only required that such an order exists.

And to what end is it required?
It cannot be done and cannot be checked. What is it good for?
And how does it "exist"?
What is ordered if there is not any property that can be named?
And where does it exist?
Questions over questions to show that the ma in matheology is of
little importance.

Your belief is tantamount to the belief in a devil or in a negative
natural number. Nothing in any rule for natural says that a natural
that cannot be used necessarily must be nonnegative. (That is only
required for naturals that can be used.)

Regards, WM

[Virgil]

In article
<4eca2790-649d-44eb...@k14g2000vbv.googlegroups.com>,

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

> On 12 Mrz., 10:05, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <aa3b230d-08a5-4e88-85a0-8b5345590...@h11g2000vbf.googlegroups.com>,
> >
> > �WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 12 Mrz., 01:24, Virgil <vir...@ligriv.com> wrote:
> >
> > > > > But in matheology, there
> > > > > are numbers that neither are nor have names.
> >
> > > > That is the case
> >
> > > And how do you put them into any order?
> >
> > Nothing in any rules for the reals says that we have to be able to, it
> > is only required that such an order exists.
>
> And to what end is it required?

For one thing, to allow us to compare the numbers we can compare.

For another, to correspond to the order properties of a line.

Certainly Cartesian Geometry is a part of even WM's limited mathematical
repertoire

> It cannot be done and cannot be checked. What is it good for?
> And how does it "exist"?

In ways clearly far beyond WM's comprehension.

>
> Your belief is tantamount to the belief in a devil or in a negative
> natural number.

In my world, 0 is the smallest natural and every other natural is
acheived by moving from there in a positive direction, and neither gods
nor devils can change that.

> Nothing in any rule for natural says that a natural
> that cannot be used necessarily must be nonnegative. (That is only
> required for naturals that can be used.)

Perhaps not in Wolkenmuekenheim but negative naturals are certainly
banned outside of it.
>
> Regards, WM







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

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.
--

[WM]

On 12 Mrz., 21:53, Virgil <vir...@ligriv.com> wrote:
> In article

> <4eca2790-649d-44eb-b5af-ae83250eb...@k14g2000vbv.googlegroups.com>,

>
>
>
>
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 12 Mrz., 10:05, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <aa3b230d-08a5-4e88-85a0-8b5345590...@h11g2000vbf.googlegroups.com>,
>
> > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > On 12 Mrz., 01:24, Virgil <vir...@ligriv.com> wrote:
>
> > > > > > But in matheology, there
> > > > > > are numbers that neither are nor have names.
>
> > > > > That is the case
>
> > > > And how do you put them into any order?
>
> > > Nothing in any rules for the reals says that we have to be able to, it
> > > is only required that such an order exists.
>
> > And to what end is it required?
>
> For one thing, to allow us to compare the numbers we can compare.

That is at most a countable set. No AC and no non-nameable numbers
required.

>
> For another, to correspond to the order properties of a line.

A line does not consist of numbers.

>
> Certainly Cartesian Geometry is a part of even WM's limited mathematical
> repertoire

Only rational points can be fixed with some precision.

>
> > It cannot be done and cannot be checked. What is it good for?
> > And how does it "exist"?
>
> In ways clearly far beyond WM's comprehension.

Similar to the understanding of the way how wonders occur and praers
are heared.

Regards, WM

[Virgil]

In article
<87cb397a-10f5-48bd...@j9g2000vbz.googlegroups.com>,

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

> On 12 Mrz., 21:53, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <4eca2790-649d-44eb-b5af-ae83250eb...@k14g2000vbv.googlegroups.com>,
> >
> >
> >
> >
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 12 Mrz., 10:05, Virgil <vir...@ligriv.com> wrote:
> > > > In article
> > > > <aa3b230d-08a5-4e88-85a0-8b5345590...@h11g2000vbf.googlegroups.com>,
> >
> > > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > > On 12 Mrz., 01:24, Virgil <vir...@ligriv.com> wrote:
> >
> > > > > > > But in matheology, there
> > > > > > > are numbers that neither are nor have names.
> >
> > > > > > That is the case
> >
> > > > > And how do you put them into any order?
> >
> > > > Nothing in any rules for the reals says that we have to be able to, it
> > > > is only required that such an order exists.
> >
> > > And to what end is it required?
> >
> > For one thing, to allow us to compare the numbers we can compare.
>
> That is at most a countable set. No AC and no non-nameable numbers
> required.

One can compare any two given decimal expansions of real number, of
which there are more than countably many possible.

> >
> > For another, to correspond to the order properties of a line.
>
> A line does not consist of numbers.

It is possible for a set of points forming a line to have more points
than the set of real numbers has numbers?

> >
> > Certainly Cartesian Geometry is a part of even WM's limited mathematical
> > repertoire
>
> Only rational points can be fixed with some precision.

WM is WRONG!
AGAIN!!
AS UUSAL!!!

On any real line in any real plane, given points 0 and 1 on thAT line,
the point corresponding to the irrational square root of 2 can be
constructed with the same theoretically perfect precision as any
rational point.

Further, the square root of many positive rationals can be constructed
with the same theoretical perfect precision as the rational itself.

While there are lots of real numbers which cannot be located on such a
real line with that perfect theoretical accuracy, not all square roots
of rationals are among them.

> >
> > > It cannot be done and cannot be checked. What is it good for?
> > > And how does it "exist"?
> >
> > In ways clearly far beyond WM's comprehension.
>
> Similar to the understanding of the way how wonders occur and praers
> are heared.

Since any geometric, and thus theoretically perfect, construction of
square roots seems to be far beyond WM's capabilities, perhaps he
should remain silent on the issue.

[WM]

On 13 Mrz., 22:32, Virgil <vir...@ligriv.com> wrote:

> > > For one thing, to allow us to compare the numbers we can compare.
>
> > That is at most a countable set. No AC and no non-nameable numbers
> > required.
>
> One can compare any two given decimal expansions of  real number, of
> which there are more than countably many possible.

No. "Given" means either a periodical expansion or defined by a finite
definition. >

> > > For another, to correspond to the order properties of a line.
>
> > A line does not consist of numbers.
>
>  It is possible for a set of points forming a line to have more points
> than the set of real numbers has numbers?

That is not the question. All numbers that can be labelled by names or
by physical points belong to a countable set.

> Further, the square root of many positive rationals can be constructed
> with the same theoretical perfect precision as the rational itself.

No. The square root can be constructed by ruler and compass with a
precision depending on the structure of the material, the grains of
sand for instance. And it can be calculated up to less than 10^100
digits. All that is not very precise and in any case rational.

Everything else is matheology.

Regards, WM

[Virgil]

In article
<605fa545-64da-44ab...@x15g2000vbj.googlegroups.com>,

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

> On 13 Mrz., 22:32, Virgil <vir...@ligriv.com> wrote:
>
> > > > For one thing, to allow us to compare the numbers we can compare.
> >
> > > That is at most a countable set. No AC and no non-nameable numbers
> > > required.
> >

> > One can compare any two given decimal expansions of real numbers, of

> > which there are more than countably many possible.
>
> No. "Given" means either a periodical expansion or defined by a finite
> definition.


> > > > For another, to correspond to the order properties of a line.
> >
> > > A line does not consist of numbers.
> >
> >  It is possible for a set of points forming a line to have more points
> > than the set of real numbers has numbers?
>
> That is not the question.

It is a reasonable question considering that you are the one who
objected to the geometry analogy.

According to standard mathematics, one can biject the set of reals to
the points on a line or on any open ended line segment.

Can one do it in your WMytheology,

> All numbers that can be labelled by names or
> by physical points belong to a countable set.

There is no such thing as an point isolatable point in physics.

>
> > Further, the square root of many positive rationals can be constructed
> > with the same theoretical perfect precision as the rational itself.
>
> No. The square root can be constructed by ruler and compass with a
> precision depending on the structure of the material, the grains of
> sand for instance. And it can be calculated up to less than 10^100
> digits. All that is not very precise and in any case rational.

Once one has the 0 to 1 interval on a line, ALL other points, including
the integer points, must be located by construction, if they can be
located at all, so none are any more actually precise, though all
constuctable points are equally theoetically precise.
>
> Everything else is matheology.

Everything outside WM's head WM regards as mathelogy, while everything
in his head is regarded by almost everyone else as pure WMytheology.

[WM]

On 14 Mrz., 02:16, Virgil <vir...@ligriv.com> wrote:
> In article

> <605fa545-64da-44ab-aa5f-a1b6652fd...@x15g2000vbj.googlegroups.com>,

>
>
>
>
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 13 Mrz., 22:32, Virgil <vir...@ligriv.com> wrote:
>
> > > > > For one thing, to allow us to compare the numbers we can compare.
>
> > > > That is at most a countable set. No AC and no non-nameable numbers
> > > > required.
>
> > > One can compare any two given decimal expansions of real numbers, of
> > > which there are more than countably many possible.
>
> > No. "Given" means either a periodical expansion or defined by a finite
> > definition.
> > > > > For another, to correspond to the order properties of a line.
>
> > > > A line does not consist of numbers.
>
> > >  It is possible for a set of points forming a line to have more points
> > > than the set of real numbers has numbers?
>
> > That is not the question.
>
>  It is a reasonable  question considering that you are the one who
> objected to the geometry analogy.
>
> According to standard mathematics, one can biject the set of reals to
> the points on a line or on any open ended line segment.

According to standar matheology one can choose one element each of an
uncountable set of sets. That is as wrong. Compare Matheology § 225.

Regards, WM

[Alan Smaill]

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

> According to standar matheology one can choose one element each of an
> uncountable set of sets. That is as wrong. Compare Matheology § 225.

You can and do of course reject this axiom.

To show something is self-contradictory, however, you need to use the
reasoning principles of the system you want to show is
self-contradictory, not your own beliefs.


> Regards, WM

--
Alan Smaill

[WM]

On 14 Mrz., 12:35, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
> WM <mueck...@rz.fh-augsburg.de> writes:
> > According to standard matheology one can choose one element each of an

> > uncountable set of sets. That is as wrong. Compare Matheology § 225.
>
> You can and do of course  reject this axiom.
>
> To show something is self-contradictory, however, you need to use the
> reasoning principles of the system you want to show is
> self-contradictory, not your own beliefs.

The axiom belongs to the system. It says that elements can be chosen.
To choose immaterial elements, hmm, how is that accomplished in a
system that contains the axiom of choice?

Regards, WM

[Alan Smaill]

I can only repeat myself --
where is the *logical* contradiction there, in terms of classical
mathematics?

Of course, you think it's false, and unimaginable, and whatever
words you want to use.

But you claim it's *self-contradictory*, don't you?

And that's a whole different claim.

[WM]

On 14 Mrz., 13:59, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 14 Mrz., 12:35, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
> >> WM <mueck...@rz.fh-augsburg.de> writes:
> >> > According to standard matheology one can choose one element each of an
> >> > uncountable set of sets. That is as wrong. Compare Matheology § 225.
>
> >> You can and do of course  reject this axiom.
>
> >> To show something is self-contradictory, however, you need to use the
> >> reasoning principles of the system you want to show is
> >> self-contradictory, not your own beliefs.
>
> > The axiom belongs to the system. It says that elements can be chosen.
> > To choose immaterial elements,  hmm, how is that accomplished in a
> > system that contains the axiom of choice?
>
> I can only repeat myself --
> where is the *logical* contradiction there, in terms of classical
> mathematics?

You will find it if you try to answer my question. Choosing means
defining (by a finite number of words) a chosen element (unless it is
a material object). No other possibility exists.

>
> Of course, you think it's false, and unimaginable, and whatever
> words you want to use.
>
> But you claim it's *self-contradictory*, don't you?
>
> And that's a whole different claim.

Please look up what Zermelo wrote. (In Matheology § 225 you will find
the orginal German text.) It is always possible /to choose/ an element
from every non-empty set and to union the chosen elements into a set
S_1.

This means: It is possible to have and to apply uncountably many
finite words in order to choose and in order to distinguish the
elements in S_1 (a set can have only distinct elements by axiom). And
the same theory says: The set of finite words is countable.
And finally: uncountable is much more than countable.

That is a contradiction.

Regards, WM