On 12/27/23 12:39 PM, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 27. Dezember 2023 um 16:22:14 UTC+1:
>
> Nonsense about quantifiers.
>
> My proof has nothing to do with quantifier shift.
>
> The proof is this: If before and after every unit fraction 1/n there is
> a distance (containing uncountably many points) up to the next unit
> fraction, then over this distance the Number of Unit Fractions between 0
> and x, NUF(x), is constant. Unless any unit fraction evades this law,
> NUF(x) must increase in steps of height 1, starting from NUF(0) = 0.
>
> Regards, WM
>

Nope.

NUF(0) = 0.

NUF(x) for any finite x > 0 is infinite, because at ANY finite point,
due to density, there are an infinite number of unit fractions before it.

The constant gap before the point x, where 1/(n+1) < x < 1/n is of
length < x/n so there is clearly room for at least n more items before
it, so there is plenty of space for more values.

There is not finite point where NUF(x) = 1, as such a point would, by
necessity, be a "First" Unit Fraction, meaning that was a "highest"
Natural Number, but, BY DEFINITION, such a thing does not exist, as ALL
Natural numbers have a successor.

So, since there can not be a "first" unit fraction, there is no finit
value where NUF(x) can have the value of 1.

So, unless YOU can find a value that evades THAT law, you arguement is
based on the assumption of something tha just doesn't exist.

If you logic requriest that there be a lowest Unit Fraction, then your
logic is just proven to be broken, and not actually based on real logic,
but just your "Dark Magic" that doesn't work/

Fritz Feldhase schrieb am Mittwoch, 27. Dezember 2023 um 16:22:14 UTC+1:
> On Wednesday, December 27, 2023 at 4:16:47 PM UTC+1, Fritz Feldhase
wrote:
> > On Wednesday, December 27, 2023 at 3:52:32 PM UTC+1, Fritz Feldhase
wrote:
> > > On Friday, December 22, 2023 at 9:30:58 AM UTC+1, WM wrote:
> > > >
> > > > "There does not exist a point of (0, 1] with less than ℵ₀
smaller unit fractions"
> > > >
> > > > is so simple to prove wrong
> > > >
> > > Oh please, Mr. Mückenheim, prove it wrong for us

With pleasure.

> > WM> There is no positive point with less than ℵo unit fractions at
its left-hand side.
> > WM> ==> All positive points with no exception have ℵo unit
fractions at their left-hand side.
> >
> > Using symbolic language: A x e IR+: E^ℵo u e UF: u < x.
> >
> > So far so good.
> >
> > WM> ==> The interval (0, 1] has aleph_0 unit fractions at its
left-hand side.
> >
> > Nope. Here you are performing a /quantifier shift/ (which is a
logical fallacy, since it can "lead" from a true statement to a false one).
> >
> > Hint: Using symbolic language you latter statement reads:

> E^ℵo u e UF: A x e (0, 1]: u < x.

No. A x e (0, 1]: E^ℵo u: u lhs x
is same as
A (x, 1]: E^ℵo u: u lhs (x, 1]
and this is, if the universal quantifier is universal,
For (0, 1]: E^ℵo u: u lhs (0, 1]
because the interval (0, 1] is nothing but all its elements and also the
limit of the sequence (x, 1].

> > But to get E^ℵo u e UF: A x e (0, 1]: u < x

That is not of interest.

Of interest is only this: If ℵo unit fractions are lhs of every x in (0,
1] but are also points of (0,1], then there must be a finite number of
unit fractions (because ℵo is the result of counting from 1) having ℵo
smaller unit fractions, which means there must be ℵo unit fractions
smaller than themselves.

Regards, WM

Le 27/12/2023 à 19:13, Richard Damon a écrit :
> On 12/27/23 11:54 AM, WM wrote:
>> Le 27/12/2023 à 14:12, Richard Damon a écrit :
>>> On 12/27/23 3:20 AM, WM wrote:
>>>> Le 26/12/2023 à 13:35, Richard Damon a écrit :
>>>>> The algorithm is DEFINED as drawing k, from the LINE of the Natural
>>>>> Numbers, and showing that this covers the MATRIX indexed by m,n.
>>>>
>>>> Precisely this happens after the integer fractions have been replaced
>>>> by the natural numbers:
>>>>
>>>> 1, 1/2, 1/3, 1/4, ...
>>>> 2, 2/2, 2/3, 2/4, ...
>>>> 3, 3/2, 3/3, 3/4, ...
>>>> 4, 4/2, 4/3, 4/4, ...
>>>> 5, 5/2, 5/3, 5/4, ...
>>>> ..
>>>>
>>>> Should this simple replacement destroy the process, then the whole
>>>> process is impossible.
>>>
>>> The k terms, of your first column, are the INPUTS,
>>
>> Of course. And everybody sees that they will never cover the whole matrix.

>
> WHo needs the INPUT to cover, the test was if the OUTPUT cover.

Whatever you wish to express. Fact is that the elements of the first
column cannot cover the whole matrix. But that is what Cantor claims.

Regards, WM

Le 27/12/2023 à 19:13, Richard Damon a écrit :
> On 12/27/23 12:39 PM, WM wrote:
>> Fritz Feldhase schrieb am Mittwoch, 27. Dezember 2023 um 16:22:14 UTC+1:
>>
>> Nonsense about quantifiers.
>>
>> My proof has nothing to do with quantifier shift.
>>
>> The proof is this: If before and after every unit fraction 1/n there is
>> a distance (containing uncountably many points) up to the next unit
>> fraction, then over this distance the Number of Unit Fractions between 0
>> and x, NUF(x), is constant. Unless any unit fraction evades this law,
>> NUF(x) must increase in steps of height 1, starting from NUF(0) = 0.
>
> NUF(0) = 0.
>
> NUF(x) for any finite x > 0 is infinite,

Your claim is disproved by the fact that every single unit fraction has a
level of constant NUF(x) behind it.

> because at ANY finite point,
> due to density, there are an infinite number of unit fractions before it.

Nonsense.
>
> The constant gap before the point x, where 1/(n+1) < x < 1/n is of
> length < x/n so there is clearly room for at least n more items before
> it, so there is plenty of space for more values.

Each one of them has a level of constant NUF(x) behind it.
>
> There is not finite point where NUF(x) = 1, as such a point would, by
> necessity, be a "First" Unit Fraction, meaning that was a "highest"
> Natural Number, but, BY DEFINITION, such a thing does not exist, as ALL
> Natural numbers have a successor.

Obviously that definition cannot be applied if all unit fractions have
gaps.
> So, unless YOU can find a value that evades THAT law, you arguement is
> based on the assumption of something tha just doesn't exist.

No it is based on soemthing that cannot be found.

Regards, WM

On 12/27/23 2:05 PM, WM wrote:
> Le 27/12/2023 à 19:13, Richard Damon a écrit :
>> On 12/27/23 12:39 PM, WM wrote:
>>> Fritz Feldhase schrieb am Mittwoch, 27. Dezember 2023 um 16:22:14 UTC+1:
>>>
>>> Nonsense about quantifiers.
>>>
>>> My proof has nothing to do with quantifier shift.
>>>
>>> The proof is this: If before and after every unit fraction 1/n there
>>> is a distance (containing uncountably many points) up to the next
>>> unit fraction, then over this distance the Number of Unit Fractions
>>> between 0 and x, NUF(x), is constant. Unless any unit fraction evades
>>> this law, NUF(x) must increase in steps of height 1, starting from
>>> NUF(0) = 0.
>>
>> NUF(0) = 0.
>>
>> NUF(x) for any finite x > 0 is infinite,
>
> Your claim is disproved by the fact that every single unit fraction has
> a level of constant NUF(x) behind it.

How does that disprove that it is infinite for all finite x. Since we
can consider "infinite" to be constant over that invertal.

>
>> because at ANY finite point, due to density, there are an infinite
>> number of unit fractions before it.
>
> Nonsense.

Maybe to you, but True.

That seems to be a core part of your problem, you don't understand how
unbounded sets work, because your logic is too primative.

>>
>> The constant gap before the point x, where 1/(n+1) < x < 1/n is of
>> length < x/n so there is clearly room for at least n more items before
>> it, so there is plenty of space for more values.
>
> Each one of them has a level of constant NUF(x) behind it.

Yep, but there is room for all of them. You can pack at least n interals
of no more than x/n length in the space of x. You don't seem to
understand basic matmematics.

>>
>> There is not finite point where NUF(x) = 1, as such a point would, by
>> necessity, be a "First" Unit Fraction, meaning that was a "highest"
>> Natural Number, but, BY DEFINITION, such a thing does not exist, as
>> ALL Natural numbers have a successor.
>
> Obviously that definition cannot be applied if all unit fractions have
> gaps.

Why not?

Where did I assume they didn't have gaps between them.

Note, 0 is NOT a "Unit Fraction" so your arguement can't force a gap
there to hide your dark numbers.

>> So, unless YOU can find a value that evades THAT law, you arguement is
>> based on the assumption of something tha just doesn't exist.
>
> No it is based on soemthing that cannot be found.

No, something that doesn't exist.

Just like your theory in the domain of truth, it just doesn't exist.

>
> Regards, WM

On 12/27/23 2:01 PM, WM wrote:
> Le 27/12/2023 à 19:13, Richard Damon a écrit :
>> On 12/27/23 11:54 AM, WM wrote:
>>> Le 27/12/2023 à 14:12, Richard Damon a écrit :
>>>> On 12/27/23 3:20 AM, WM wrote:
>>>>> Le 26/12/2023 à 13:35, Richard Damon a écrit :
>>>>>> The algorithm is DEFINED as drawing k, from the LINE of the
>>>>>> Natural Numbers, and showing that this covers the MATRIX indexed
>>>>>> by m,n.
>>>>>
>>>>> Precisely this happens after the integer fractions have been
>>>>> replaced by the natural numbers:
>>>>>
>>>>> 1, 1/2, 1/3, 1/4, ...
>>>>> 2, 2/2, 2/3, 2/4, ...
>>>>> 3, 3/2, 3/3, 3/4, ...
>>>>> 4, 4/2, 4/3, 4/4, ...
>>>>> 5, 5/2, 5/3, 5/4, ...
>>>>> ..
>>>>>
>>>>> Should this simple replacement destroy the process, then the whole
>>>>> process is impossible.
>>>>
>>>> The k terms, of your first column, are the INPUTS,
>>>
>>> Of course. And everybody sees that they will never cover the whole
>>> matrix.
>
>>
>> WHo needs the INPUT to cover, the test was if the OUTPUT cover.
>
> Whatever you wish to express. Fact is that the elements of the first
> column cannot cover the whole matrix. But that is what Cantor claims.
>
> Regards, WM

No, he never mapped "Matrix B to Matrix B", but the Natural Numbers to
the pairs m,n.

You just don't understand what he is doing, becaue your knowledge is too
limited.

Le 27/12/2023 à 20:34, Richard Damon a écrit :
> On 12/27/23 2:05 PM, WM wrote:

>>> NUF(0) = 0.
>>>
>>> NUF(x) for any finite x > 0 is infinite,
>>
>> Your claim is disproved by the fact that every single unit fraction has
>> a level of constant NUF(x) behind it.
>
> How does that disprove that it is infinite for all finite x. Since we
> can consider "infinite" to be constant over that invertal.

The increase from 0 to more happens via unit fractions which all have a
gap behind them.
It is impossible that NUF can increase infinitely anywhere without being
constant in between.

Regards, WM

Le 27/12/2023 à 20:34, Richard Damon a écrit :
> On 12/27/23 2:01 PM, WM wrote:

>> Whatever you wish to express. Fact is that the elements of the first
>> column cannot cover the whole matrix. But that is what Cantor claims.

>
> No, he never mapped "Matrix B to Matrix B", but the Natural Numbers to
> the pairs m,n.

But there are to few natural numbers. There are to few indices X in

XOOO...
XOOO...
XOOO...
XOOO...
...

cover the whole matrix.

Regards, WM

On 12/27/23 4:47 PM, WM wrote:
> Le 27/12/2023 à 20:34, Richard Damon a écrit :
>> On 12/27/23 2:05 PM, WM wrote:
>
>>>> NUF(0) = 0.
>>>>
>>>> NUF(x) for any finite x > 0 is infinite,
>>>
>>> Your claim is disproved by the fact that every single unit fraction
>>> has a level of constant NUF(x) behind it.
>>
>> How does that disprove that it is infinite for all finite x. Since we
>> can consider "infinite" to be constant over that invertal.
>
> The increase from 0 to more happens via unit fractions which all have a
> gap behind them.
> It is impossible that NUF can increase infinitely anywhere without being
> constant in between.
>
> Regards, WM

But there isn't a "gap" between 0 and the unbounded set of unit
fractions, so you "set theory" can't actually say anything about what
happens there.

It is "Dark" about that point, so provides no basis.

Is can't increase infinitely anywhere BETWEEN UNIT FRACTIONS, but
between 0 and the positive numbers, it isn't defined, so can arbitrarily
junp, in fact it MUST because the Unit Fractions get unboundly small so
there can't be a "first" unit fraction for NUF to be 1 at.

It can't even be a "dark" Unit fraction, if dark unit fractions are a
part of the set of unit fractions, as even a dark unit fraction would
have unit fractions smaller than it.

At best, your "dark" numbers that NUF(x) increase in finite values would
be numbers in the infinitisimals, but you specificially say that ish't
what you are talking about, thus you have shown that NUF(x) CAN'T be 1
at any value of x. Until you can show some theory that actually defines
this number and shows that it IS a "Unit Fraction", being the reciprocal
of a Natural Number, all you have done is show that you don't know what
you are talking about.

You use reasoning from insufficiently strong logic system to actually
generate the Natural Numbers, that just proves that your system, if it
tries to work with Natural Numbers breaks.

*BOOM*

On 12/27/23 4:50 PM, WM wrote:
> Le 27/12/2023 à 20:34, Richard Damon a écrit :
>> On 12/27/23 2:01 PM, WM wrote:
>
>>> Whatever you wish to express. Fact is that the elements of the first
>>> column cannot cover the whole matrix. But that is what Cantor claims.
>
>>
>> No, he never mapped "Matrix B to Matrix B", but the Natural Numbers to
>> the pairs m,n.
>
> But there are to few natural numbers. There are to few indices X in
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ..
>
> cover the whole matrix.
>
> Regards, WM
>
>

But there is not, when you transform k -> m,n where

k = = (m + n - 1)(m + n - 2)/2 + m

you find there is exactly one combination of m,n for every k, so there
are just as many k's as m,n combinations.

Thus, we have found a valid bijection.

This shows that Aleph0 squared is still Aleph0, and in fact, Aleph0 to
any finite power is still Aleph0.

(you don't get to Aleph1 until you start taking things to teh Aleph0
power, and that becomes uncountable)

Le 28/12/2023 à 03:35, Richard Damon a écrit :

> But there isn't a "gap" between 0 and the unbounded set of unit
> fractions, so you "set theory" can't actually say anything about what
> happens there.
>
> Is can't increase infinitely anywhere BETWEEN UNIT FRACTIONS,

Correct. Therefore the set cannot exist without gap. At most one unit
fraction can.

> but
> between 0 and the positive numbers, it isn't defined

incorrect.

, so can arbitrarily
> junp, in fact it MUST because the Unit Fractions get unboundly small so
> there can't be a "first" unit fraction for NUF to be 1 at.

ZFC supplies for the function Number of Unit Fractions between 0 and x the
result
NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
So, according to ZFC the number of unit fractions grows between 0 and (0,
1] by more than 2.
This is impossible because between any two unit fractions there are
"uncountably" many points which do not fit between 0 and (0, 1]. This
proves that ZFC does not deliver correct mathematics in this case, like in
several others:
https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory

Regards, WM

Le 28/12/2023 à 03:35, Richard Damon a écrit :
> On 12/27/23 4:50 PM, WM wrote:

>> But there are too few natural numbers. There are too few indices X in
>>
>> XOOO...
>> XOOO...
>> XOOO...
>> XOOO...
>> ..
>>
>> cover the whole matrix.
>
> But there is not, when you transform k -> m,n where
>
> k = = (m + n - 1)(m + n - 2)/2 + m
>
> you find there is exactly one combination of m,n for every k, so there
> are just as many k's as m,n combinations.

Hence there is an internal contradiction.
>
> Thus, we have found a valid bijection.

Together with the above matrix: an interbal contradiction.

Regards, WM

Fritz Feldhase schrieb am Mittwoch, 27. Dezember 2023 um 23:47:40 UTC+1:
> On Wednesday, December 27, 2023 at 7:30:52 PM UTC+1, WM wrote:

> > A x e (0, 1]: E^ℵo u: u < x
> > is same as
> > A (x, 1]: E^ℵo u: u < (x, 1]
>
> Huh?! "A (x, 1]: E^ℵo u: u < (x, 1]" ist noch nicht mal eine wff,

Better correct mathematics than well formed credo in absurdum.

ZFC supplies for the function Number of Unit Fractions between 0 and x
the result
NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
So, according to ZFC the number of unit fractions grows between 0 and
(0, 1] by more than 2.
This is impossible because between any two unit fractions there are
"uncountably" many points which do not fit between 0 and (0, 1]. This
proves that ZFC does not deliver correct mathematics in this case, like
in several others:
https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory

Regards, WM

On 12/28/23 3:44 AM, WM wrote:
> Le 28/12/2023 à 03:35, Richard Damon a écrit :
>> On 12/27/23 4:50 PM, WM wrote:
>
>>> But there are too few natural numbers. There are too few indices X in
>>>
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> ..
>>>
>>> cover the whole matrix.
>>
>> But there is not, when you transform k -> m,n where
>>
>> k = = (m + n - 1)(m + n - 2)/2 + m
>>
>> you find there is exactly one combination of m,n for every k, so there
>> are just as many k's as m,n combinations.
>
> Hence there is an internal contradiction.

WHAT is in contradiction?

He talks NOTHING of "taking from a column", that is YOUR invention.

>>
>> Thus, we have found a valid bijection.
>
> Together with the above matrix: an interbal  contradiction.
>
> Regards, WM
>
>
>

The only contradiction is trying to resolve what you say with having
intelligence.

On 12/28/23 3:42 AM, WM wrote:
> Le 28/12/2023 à 03:35, Richard Damon a écrit :
>
>> But there isn't a "gap" between 0 and the unbounded set of unit
>> fractions, so you "set theory" can't actually say anything about what
>> happens there.
>>
>> Is can't increase infinitely anywhere BETWEEN UNIT FRACTIONS,
>
> Correct. Therefore the set cannot exist without gap. At most one unit
> fraction can.

Which says NOTHING about between 0 and the set, since 0 isn't a "Unit
Fraction"

>
>> but between 0 and the positive numbers, it isn't defined
>
> incorrect.

Show the definition?

>
> , so can arbitrarily
>> junp, in fact it MUST because the Unit Fractions get unboundly small
>> so there can't be a "first" unit fraction for NUF to be 1 at.
>
> ZFC supplies for the function Number of Unit Fractions between 0 and x
> the result
> NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
> So, according to ZFC the number of unit fractions grows between 0 and
> (0, 1] by more than 2.
> This is impossible because between any two unit fractions there are
> "uncountably" many points which do not fit between 0 and (0, 1]. This
> proves that ZFC does not deliver correct mathematics in this case, like
> in several others:
> https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory
>
> Regards, WM
>

You do understand that ZFC does not provide a full definition of the
"Natural Numbers", and never claims to? (It can provide a close
approximation, but not the full properties of them). My understanding is
that this is, in part, due to ZFC being a "First Order" logic system,
and the full properties of the Natural Numbers needing Second Order logic.

Thus, trying to use it in a system that DOES have a full definition of
the Natural Numbers leads you to inconsistencies. And the fault there is
YOURS, not the theory.

Perhaps you should read your set theory a bit more closely.

Your logic is built on the LIE of assuming there must be a first unit
fraction. Without that lie, you don't get your contradiction, just the
fact that your NUF function never has a finite value other than 0, as to
have a value of 1, there must be a first unit fraction, which there is not.

On 12/28/23 3:50 AM, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 27. Dezember 2023 um 23:47:40 UTC+1:
> > On Wednesday, December 27, 2023 at 7:30:52 PM UTC+1, WM wrote:
>
> > > A x e (0, 1]: E^ℵo u: u < x
> > > is same as
> > > A (x, 1]: E^ℵo u: u < (x, 1]
> >
> > Huh?! "A (x, 1]: E^ℵo u: u < (x, 1]" ist noch nicht mal eine wff,
>
> Better correct mathematics than well formed credo in absurdum.
>
> ZFC supplies for the function Number of Unit Fractions between 0 and x
> the result
> NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
> So, according to ZFC the number of unit fractions grows between 0 and
> (0, 1] by more than 2.
> This is impossible because between any two unit fractions there are
> "uncountably" many points which do not fit between 0 and (0, 1]. This
> proves that ZFC does not deliver correct mathematics in this case, like
> in several others:
> https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory
>
> Regards, WM
>

And ZFC can't be used here, as you are using a set (The actual Natural
Numbers, with all their properties). It can generate the values
themselfs as the set omega, but not all their properties.

Show how in ZFC you can actually derive your function "NUF(x)", using
only the tools defined in ZFC, and not assumed from other theories of
the Natural Numbers.

Oh, and there ARE "uncountably" many points between 0 and (0, 1], if you
are in the right number system, as the infinitesimals fit in there.

But, you reject those, but still use their properties.

Le 28/12/2023 à 14:07, Richard Damon a écrit :
> On 12/28/23 3:44 AM, WM wrote:
>> Le 28/12/2023 à 03:35, Richard Damon a écrit :
>>> On 12/27/23 4:50 PM, WM wrote:
>>
>>>> But there are too few natural numbers. There are too few indices X in
>>>>
>>>> XOOO...
>>>> XOOO...
>>>> XOOO...
>>>> XOOO...
>>>> ..
>>>>
>>>> cover the whole matrix.
>>>
>>> But there is not, when you transform k -> m,n where
>>>
>>> k = = (m + n - 1)(m + n - 2)/2 + m
>>>
>>> you find there is exactly one combination of m,n for every k, so there
>>> are just as many k's as m,n combinations.
>>
>> Hence there is an internal contradiction.
>
> WHAT is in contradiction?

XOOO...
XOOO...
XOOO...
XOOO...
...

> He talks NOTHING of "taking from a column", that is YOUR invention.

He does not even talk about a matrix. But he talks about a series (older
form of sequence) an that applies all natural numbers as indices of the
terms. They are the integer fractions, i.e., the elements of the first
column. You need not even much intelligence to understand that - but a
bit.

Regards, WM

Le 28/12/2023 à 14:07, Richard Damon a écrit :
> On 12/28/23 3:42 AM, WM wrote:

>> ZFC supplies for the function Number of Unit Fractions between 0 and x
>> the result
>> NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
>> So, according to ZFC the number of unit fractions grows between 0 and
>> (0, 1] by more than 2.
>> This is impossible because between any two unit fractions there are
>> "uncountably" many points which do not fit between 0 and (0, 1]. This
>> proves that ZFC does not deliver correct mathematics in this case, like
>> in several others:
>>
>> https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory
>>
> You do understand that ZFC does not provide a full definition of the
> "Natural Numbers",

I understand that it is the case, but hitherto I haven't found a
matheologian who confessed that.

> and never claims to?

That is new to me. ZFC is the theory of all sets, in particular of the
smallest infinite set.

> Thus, trying to use it in a system that DOES have a full definition of
> the Natural Numbers leads you to inconsistencies.

Can you give a reference for this opinion? Is there a "professional"
mathematician who confessed this? Yesterday a student of mine has asked
the same question in MathOverflow. But five dishonest fools there have
quickly deleted his question, as not matching their level of stupidity, of
course without even trying to answer it. Therefore I am interested to find
out whether there are honest matheologians somewhere in the world.

Regards, WM

On 12/28/23 11:11 AM, WM wrote:
> Le 28/12/2023 à 14:07, Richard Damon a écrit :
>> On 12/28/23 3:44 AM, WM wrote:
>>> Le 28/12/2023 à 03:35, Richard Damon a écrit :
>>>> On 12/27/23 4:50 PM, WM wrote:
>>>
>>>>> But there are too few natural numbers. There are too few indices X in
>>>>>
>>>>> XOOO...
>>>>> XOOO...
>>>>> XOOO...
>>>>> XOOO...
>>>>> ..
>>>>>
>>>>> cover the whole matrix.
>>>>
>>>> But there is not, when you transform k -> m,n where
>>>>
>>>> k = = (m + n - 1)(m + n - 2)/2 + m
>>>>
>>>> you find there is exactly one combination of m,n for every k, so
>>>> there are just as many k's as m,n combinations.
>>>
>>> Hence there is an internal contradiction.
>>
>> WHAT is in contradiction?
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ..

And what is the contradiction about that?

Who expects that one column of a matrix will ever fill the full area of
the SAME matrix?

If you do, your logic is just broken.

>
>> He talks NOTHING of "taking from a column", that is YOUR invention.
>
> He does not even talk about a matrix. But he talks about a series (older
> form of sequence) an that applies all natural numbers as indices of the
> terms. They are the integer fractions, i.e., the elements of the first
> column. You need not even much intelligence to understand that - but a bit.
>
> Regards, WM
>
>

Right, he shows a mapping of the "1 dimensional" sequence of k, to the
"2 dimensional" sequence set of m,n. We can display the 2-dimensional
sequence in a matrix.

So, he does what he claims, produces a mapping of the Counting Numbers
to fill the space of the number pairs.

YOU are the one that somehow thinks this should fill the number pairs
out of the set of just the Natural Numbers, which shows your ignorance
of what you are actually trying to do.

On 12/28/23 11:28 AM, WM wrote:
> Le 28/12/2023 à 14:07, Richard Damon a écrit :
>> On 12/28/23 3:42 AM, WM wrote:
>
>>> ZFC supplies for the function Number of Unit Fractions between 0 and
>>> x the result
>>> NUF(x) = 0 for x =< 0 and NUF(x) = ℵo for x > 0.
>>> So, according to ZFC the number of unit fractions grows between 0 and
>>> (0, 1] by more than 2.
>>> This is impossible because between any two unit fractions there are
>>> "uncountably" many points which do not fit between 0 and (0, 1]. This
>>> proves that ZFC does not deliver correct mathematics in this case,
>>> like in several others:
>>> https://www.researchgate.net/publication/376587391_The_seven_deadly_sins_of_set_theory
>>>
>> You do understand that ZFC does not provide a full definition of the
>> "Natural Numbers",
>
> I understand that it is the case, but hitherto I haven't found a
> matheologian who confessed that.

Yes, it creates the SET of the Natural Numbers, but it does not create
the full MATHEMATICS of the Natural Numbers.

>
>> and never claims to?
>
> That is new to me. ZFC is the theory of all sets, in particular of the
> smallest infinite set.

As I said, it creates the SET of the NUMBERS, but doesn't create the
full mathematics of the natural Numbers.

>> Thus, trying to use it in a system that DOES have a full definition of
>> the Natural Numbers leads you to inconsistencies.
>
> Can you give a reference for this opinion? Is there a "professional"
> mathematician who confessed this? Yesterday a student of mine has asked
> the same question in MathOverflow. But five dishonest fools there have
> quickly deleted his question, as not matching their level of stupidity,
> of course without even trying to answer it. Therefore I am interested to
> find out whether there are honest matheologians somewhere in the world.
>
> Regards, WM
>

Just read Godel. He shows that any system that creates the full (or
actually, just enough of the) mathematics of the Natural Numbers must be
incomplete, but also points out the ZFC can be complete but not be able
to prove it consistency. This is because ZFC doesn't generate the full
mathematics.

This is because ZFC just uses First Order Logic, which isn't enough to
create the full mathematics, (but is enough to generate the set of numbers).

The Mathematics requries adding at least induction, which is a second
order logical form, and thus beyond basic ZFC.

As to formal references, I will admit this isn't my field of expertise,
and thus I don't know the formal source material that lays this out. I
am really an "Engineer" that uses this sort of Theory to make things
work, and thus have an understanding of the basic principles.

On Wednesday, December 27, 2023 at 4:24:30 AM UTC-4, WM wrote:
> My game shows that the matrix A
Your game does not provide a definition for the putative A

Le 28/12/2023 à 19:34, Richard Damon a écrit :
> On 12/28/23 11:11 AM, WM wrote:

>>>
>>> WHAT is in contradiction?
>>
>> XOOO...
>> XOOO...
>> XOOO...
>> XOOO...
>> ..
>
> And what is the contradiction about that?
>
> Who expects that one column of a matrix will ever fill the full area of
> the SAME matrix?

Every set theorist does.
>
> If you do, your logic is just broken.

I don't. But Cantor claims that all positive unit fractions can be
enumerated by all natural numbers. All positive unit fractions are the
positions of the matrix. All integer fractions in the first column supply
all natural numbers. There can be mo objection to that.

> Right, he shows a mapping of the "1 dimensional" sequence of k, to the
> "2 dimensional" sequence set of m,n. We can display the 2-dimensional
> sequence in a matrix

and the one-dimensional sequence in its first column.

Regards, WM

Le 28/12/2023 à 19:34, Richard Damon a écrit :
> On 12/28/23 11:28 AM, WM wrote:

>> Can you give a reference for this opinion? Is there a "professional"
>> mathematician who confessed this?
>
> Just read Godel. He shows that any system that creates the full (or
> actually, just enough of the) mathematics of the Natural Numbers must be
> incomplete, but also points out the ZFC can be complete but not be able
> to prove it consistency. This is because ZFC doesn't generate the full
> mathematics.

And this should prevent the function NUF(x) to be observed completely?
>
> This is because ZFC just uses First Order Logic, which isn't enough to
> create the full mathematics, (but is enough to generate the set of numbers).

This distinction is simply nonsense.There is logic and there is unlogic.
That's all. But anyhow NUF(x) is first order logic.
>
> The Mathematics requries adding at least induction, which is a second
> order logical form, and thus beyond basic ZFC.

To my knowledge, no set theorist does agree with you.
>
> As to formal references, I will admit this isn't my field of expertise,
> and thus I don't know the formal source material that lays this out. I
> am really an "Engineer" that uses this sort of Theory to make things
> work, and thus have an understanding of the basic principles.

An engineer who really would apply set theory and mathematical "logic" is
doomed to failure.

Regards, WM

On 12/30/23 6:21 AM, WM wrote:
> Le 28/12/2023 à 19:34, Richard Damon a écrit :
>> On 12/28/23 11:28 AM, WM wrote:
>
>>> Can you give a reference for this opinion? Is there a "professional"
>>> mathematician who confessed this?
>>
>> Just read Godel. He shows that any system that creates the full (or
>> actually, just enough of the) mathematics of the Natural Numbers must
>> be incomplete, but also points out the ZFC can be complete but not be
>> able to prove it consistency. This is because ZFC doesn't generate the
>> full mathematics.
>
> And this should prevent the function NUF(x) to be observed completely?

The problem with NUF is in its definition.

>>
>> This is because ZFC just uses First Order Logic, which isn't enough to
>> create the full mathematics, (but is enough to generate the set of
>> numbers).
>
> This distinction is simply nonsense.There is logic and there is unlogic.
> That's all. But anyhow NUF(x) is first order logic.

No, it is UNLOGIC, at least the assertion that a point exists where
NUF(x) == 1. Please try to show the ACTUAL LOGIC (not some "dark logic")
that shows that there exists a value of x that NUF(x) is 1 at.

Note, the fact that at the point x in (0, 1], 1/(n+1) < x <= 1/n, there
is a flat point doesn't prove what you want. The fact that the gap it is
in is only 1/((n*(n+1)) means the gap is much SMALLER than x, and can
never reach to 0, so there does not exist a value of x where the gap
before it can be long enough to reach to 0, and thus no point where
NUF(x) can be 1.

You are trying to argue that the gaps somehow "fill" the space and force
you to squeeze a point to have NUF(x) == 1, but in fact, because the
flat area has a maximum length at any given point, it shows that this
can't happen.

You are just ignoring the parts that seem inconvient to you.

>>
>> The Mathematics requries adding at least induction, which is a second
>> order logical form, and thus beyond basic ZFC.
>
> To my knowledge, no set theorist does agree with you.

Since you think k <-> 1/k is the same mapping as
k <-> m/n with k = (m+n-1)(m+n-2)/2 + m

Your knowledge doesn't really matter, facts matter.

>>
>> As to formal references, I will admit this isn't my field of
>> expertise, and thus I don't know the formal source material that lays
>> this out. I am really an "Engineer" that uses this sort of Theory to
>> make things work, and thus have an understanding of the basic principles.
>
> An engineer who really would apply set theory and mathematical "logic"
> is doomed to failure.
> Regards, WM
>
>

Someone who thinks there must be a highest Natural Number (or smallest
unit fraction) has already failed.

On 28.12.2023 21:39, William wrote:
> On Wednesday, December 27, 2023 at 4:24:30 AM UTC-4, WM wrote:
>> My game shows that the matrix A is
larger than B and containing not indexed fractions.

But whether or not A is defined, the matrix
XOOO...
XOOO...
XOOO...
XOOO...
....
will never be covered by X. This shows that never all matrix positions
will be indeXed.

> Your game does not provide a definition for the putative A

That is because most of it is dark. By the way the same is true for B.

Regards, WM