On Saturday, January 28, 2023 at 3:43:08 PM UTC-4, WM wrote:
> William schrieb am Samstag, 28. Januar 2023 um 20:19:32 UTC+1:
> > On Saturday, January 28, 2023 at 1:50:45 PM UTC-4, WM wrote:
>
> > > > Incorrect. For any real 0<x<1 the interval (x,1] contains only a finite number of elements of U_P.
> > > Yes, that is true. But why do you say incorrect? Can you discern more U_P?
> > Yes, Every subset of U_P of cardinality 1 has the property that it can be written down
> No. All subsets of cardinality 1 which belong to the difference*) cannot be written down.

Incorrect. There is no subset of U_P of cardinality 1 (whether it belongs do the difference or not) which cannot be written down. (Note this does not imply, '"The subset of U_P which belongs to the difference can be written down"). U_P is a Peano set. It has cadinality aleph_0

JVR schrieb am Samstag, 28. Januar 2023 um 21:23:03 UTC+1:
> On Saturday, January 28, 2023 at 6:53:27 PM UTC+1, WM wrote:
> > JVR schrieb am Samstag, 28. Januar 2023 um 12:46:33 UTC+1:
> > > On Saturday, January 28, 2023 at 12:36:05 PM UTC+1, WM wrote:
> >
> > > > > > Would you be so kind to note which step is wrong in your opinion?
> > > > > 2, 3, and 4 are the usual meaningless garble.
> > > > > Define your terms.
> > > > Define a unit fraction such that the difference disappears. Then you will learn what defining means.
> > > >
> > > Almost nobody will notice.
> >
> > You err. My PDF Dark numbers II in Research Gate has by far more than 2000 reads.
> >
> Yes, I hear that the AMS is about to fire all it's editors because they refused to publish your
> fascinating research papers.

I have never submitted a paper to AMS. I have never submiited anything about dark numbers. Printed papers are outdated.

Regards, WM

William schrieb am Samstag, 28. Januar 2023 um 21:37:04 UTC+1:
> On Saturday, January 28, 2023 at 3:43:08 PM UTC-4, WM wrote:
> > William schrieb am Samstag, 28. Januar 2023 um 20:19:32 UTC+1:
> > > On Saturday, January 28, 2023 at 1:50:45 PM UTC-4, WM wrote:
> >
> > > > > Incorrect. For any real 0<x<1 the interval (x,1] contains only a finite number of elements of U_P.
> > > > Yes, that is true. But why do you say incorrect? Can you discern more U_P?
> > > Yes, Every subset of U_P of cardinality 1 has the property that it can be written down
> > No. All subsets of cardinality 1 which belong to the difference*) cannot be written down.
> Incorrect. There is no subset of U_P of cardinality 1 (whether it belongs do the difference or not) which cannot be written down.

There are many single unit fractions 1/n which are subsets {1/n} of cardinality 1.

> (Note this does not imply, '"The subset of U_P which belongs to the difference can be written down").

The subset can be written down. It is the difference set. Note that I wrote it: "difference set"
But its elements cannot be written down.

Regards, WM

Crank Wolfgang Mückenheim, aka WM wrote:
> ...
> There are many single unit fractions 1/n which are subsets {1/n} of cardinality 1.

Here Crank Wolfgang Mückenheim, from Hochschule Augsburg, *explicitely*
use the void argument that everything is a member of a finite set.
Go figure: a is a member of {a} whatever a could be... He did that
a lot of times, without properly stating clearly how absurdly void
such a statement is...

Now Crank Wolfgang Mückenheim, from Hochschule Augsburg, is actually
*explicitely* stating down such an argument...

You are a shame, Mückenheim, that you've been allowed to teach for
years is a absolute disgrace for German Academy.

Python schrieb am Samstag, 28. Januar 2023 um 23:37:10 UTC+1:

Nonsense.

Every unit fraction is a subset of cardinality 1 of the infinite set of all unit fractions.

Regards, WM

Python wrote:
> Crank Wolfgang Mückenheim, aka WM wrote:
>> ...
>> There are many single unit fractions 1/n which are subsets {1/n} of
>> cardinality 1.
>
> Here Crank Wolfgang Mückenheim, from Hochschule Augsburg, *explicitely*
> use the void argument that everything is a member of a finite set.
> Go figure: a is a member of {a} whatever a could be... He did that
> a lot of times, without properly stating clearly how absurdly void
> such a statement is...
>
> Now Crank Wolfgang Mückenheim, from Hochschule Augsburg, is actually
> *explicitely* stating down such an argument...
>
> You are a shame, Mückenheim, that you've been allowed to teach for
> years is a absolute disgrace for German Academy.

Moreover he did it wrongly, LOL!

>>> There are many single unit fractions 1/n which are subsets {1/n} of
>>> cardinality 1.

1/n is not a subset of {1/n}

Crank Wolfgang Mückenheim, from Hochschule Augsburg, is a fractal
of errors. It ends out that there are almost NOT a single page
without a damming error in his infamous book about elementary
math... What the Hell is happening in Germany to let such crap
book to be published and cranky "professor" to teach?

Python schrieb am Samstag, 28. Januar 2023 um 23:46:18 UTC+1:

> >>> There are many single unit fractions 1/n which are subsets {1/n} of
> >>> cardinality 1.
> 1/n is not a subset of {1/n}

I said above that {1/n} is a subset of the set of all unit fractions.
Can't you read?
Every unit fraction is a subset of cardinality 1 of the infinite set of all unit fractions.

Are you a professor? Where?

Regards, WM

Crank Wolfgang Mückenheim, aka WM wrote:
> Python schrieb am Samstag, 28. Januar 2023 um 23:37:10 UTC+1:
>
> Nonsense.
>
> Every unit fraction is a subset of cardinality 1 of the infinite set of all unit fractions.

You are an idiot, Crank Wolfgang Mückenheim, from Hochschule Augsburg.

A member of a set is (in general) not a subset of this set.

1/2 is definitely NOT a subset of {1/2}

You are a shame, Mückenheim, that you've been allowed to teach for
years is a absolute disgrace for German Academy.

On Saturday, January 28, 2023 at 6:30:02 PM UTC-4, WM wrote:
> William schrieb am Samstag, 28. Januar 2023 um 21:37:04 UTC+1:
> > On Saturday, January 28, 2023 at 3:43:08 PM UTC-4, WM wrote:
> > > William schrieb am Samstag, 28. Januar 2023 um 20:19:32 UTC+1:
> > > > On Saturday, January 28, 2023 at 1:50:45 PM UTC-4, WM wrote:
> > >
> > > > > > Incorrect. For any real 0<x<1 the interval (x,1] contains only a finite number of elements of U_P.
> > > > > Yes, that is true. But why do you say incorrect? Can you discern more U_P?
> > > > Yes, Every subset of U_P of cardinality 1 has the property that it can be written down
> > > No. All subsets of cardinality 1 which belong to the difference*) cannot be written down.
> > Incorrect. There is no subset of U_P of cardinality 1 (whether it belongs do the difference or not) which cannot be written down.
> There are many single unit fractions 1/n which are subsets {1/n} of cardinality 1.

Indeed. U_P is a subset of {1/n}. U_P is a Peano set. There are an infinite number of *elements* of U_P. Each is finite.. There is no element that cannot be written down. The *set* U_P ("All elements of U_P") is infinite and cannot be written down.

Crank Wolfgang Mückenheim, aka WM wrote:
> Python schrieb am Samstag, 28. Januar 2023 um 23:46:18 UTC+1:
>
>>>>> There are many single unit fractions 1/n which are subsets {1/n} of
>>>>> cardinality 1.
>> 1/n is not a subset of {1/n}
>
> I said above that {1/n} is a subset of the set of all unit fractions.
> Can't you read?
> Every unit fraction is a subset of cardinality 1 of the infinite set of all unit fractions.

No you wrote that a single unit fraction 1/n is a subset of cardinality
1 (while actually it a equivalence class, so a infinite set of pairs
of natural number: {(a,b) | a*n = b}).

Everything is a member of a finite set, you goofed by confusing members
and subsets, only someone as demented as you can pretend to draw some
conclusion from that.

> Are you a professor? Where?

In places where students are not abused and molested by people of your
kind, Crank Wolfgang Mückenheim, from Hochshule Augsburg.

On 1/28/2023 12:44 PM, WM wrote:
> Jim Burns schrieb am Samstag,
> 28. Januar 2023 um 15:36:02 UTC+1:
>> On 1/28/2023 6:11 AM, WM wrote:
>>> Jim Burns schrieb am Samstag,
>>> 28. Januar 2023 um 00:33:43 UTC+1:

>>>> You're asking
>>>> why is there no interval with
>>>> finitely-many rationals in it?
>>>
>>> Yes.
>>
>> Because
>
> Your explanation is batantly wrong.
> Proof:
> you cannot keep Bob and you cannot delete Bob.

Yes,
you cannot keep Bob and you cannot delete Bob.

That is closely related to the rationals
in one open interval (a,b) being matchable to
the rationals in any other open interval (c,d)
of rationals.

Let Bob be a rational in (1/2,1)
The rationals in (0,1/2) can be matched to
the rationals in (0,1)

After all the swaps p/2:p of rationals in
(0,1/2) and (0,1), Bob is not deleted and
Bob is not kept.
'Bye, Bob.

Bob-conservation is a property of finite sets.

(0,1/2), (0,1), (a,b), (c,d) are not finite
sets.

It is also why there are no intervals with
finitely-many rationals in it,
which is why any movement along the real line
is impossible if only finitely-many points are
permitted to be crossed.

> That is a contradiction.

It is a contradiction of the assumption that
(0,1/2), (0,1), (a,b), (c,d) have Bob-conservation.

Bob-conservation is a property of finite sets.
Infinity is not
a reallyreallyreallyreallyreallyreally large
natural number.

> But it is irrelevant.
>
>>>>> You said that ℵ₀ points are passed
>>>>> before you can stop. Do they exist?
>>>>
>>>> Let q be an element of ℚ⁺
>>>
>>> No, the question was: Do they exist?
>>
>> Suppose I describe some points.
>
> You cannot describe more than finitely many.

I can make a claim which is not-possibly-false
about about each element of ℚ⁺
even if there are infinitely-many,
even if there is no way to know
which element of ℚ⁺ is referred to.

I can augment that claim with only visibly
not-first-possibly-false claims about
an element of ℚ⁺

Each claim in the sequence is not-first-
-possibly-false and about an element of ℚ⁺

Therefore,
each claim in the sequence is
not-possibly-false about an element of ℚ⁺
even if there are infinitely-many,
even if there is no way to know
which element of ℚ⁺ is referred to.

On Sunday, 8 January 2023 at 10:35:15 UTC+2, WM wrote:
> "It never happens that the (always finite number of) pathbunches split into 2^ℵo pathbunches. However, if we posit a time after an infinite number of steps, we can say that the split happened. This 'it never happens but it happened' is unique to processes with no last step. For a process with no last step, you cannot conclude something cannot have happened, because there is no step where it happens."
>
> "You are apparently trying to appeal to a theorem that says that if a_n < A for all n then lim a_n =< A. This statement is false in the context;"
>
> "At every level we have more lamettas than nodes in the binary tree. But through each of these nodes pass infinitely many paths. Maybe you can not distinguish them just by looking at the nodes but they do exist."
>
> "Unfortunately you have not provided a way of moving this from 'true for all finite' to 'true for infinite n', and induction won't take you from 'all finite n' to 'infinite n'."
>
> "If all nodes of a path have been deleted, that does not mean you ever deleted the path!!! You could delete all and only the finite paths from the tree, and that would delete all the nodes, yet you would never have deleted any infinite path, and they would all still exist in any case, regardless of what had been deleted!"
>
> Gruß, WM

Like I told you many moons ago, you will lose any argument because you do not understand that the concept of NUMBER is at the core of all these arguments.

A NUMBER is a NAME given to a MEASURE of a RATIO of MAGNITUDES (NOT NUMBERS!)

The very first line of argument "2^ℵo pathbunches" is already in error since ℵo is NOT a NUMBER.

https://www.academia.edu/89794682/Understanding_the_concept_of_number_in_just_2_pages

Jim Burns schrieb am Sonntag, 29. Januar 2023 um 00:30:35 UTC+1:
> On 1/28/2023 12:44 PM, WM wrote:

> > you cannot keep Bob and you cannot delete Bob.
> Yes,
> you cannot keep Bob and you cannot delete Bob.
> That is closely related to the rationals
> in one open interval (a,b) being matchable to
> the rationals in any other open interval (c,d)
> of rationals.

That would be the proof that "at infinity" everything may happen. No mathematics based upon two-valued logic possible. All your arguing would be in vain.

Regards, WM

On 1/29/2023 1:29 PM, WM wrote:
> Jim Burns schrieb am Sonntag,
> 29. Januar 2023 um 00:30:35 UTC+1:
>> On 1/28/2023 12:44 PM, WM wrote:

>>> you cannot keep Bob and you cannot delete Bob.
>>
>> Yes,
>> you cannot keep Bob and you cannot delete Bob.
>> That is closely related to the rationals
>> in one open interval (a,b) being matchable to
>> the rationals in any other open interval (c,d)
>> of rationals.
>
> That would be the proof that "at infinity"
> everything may happen.

No,
it is a proof that
some sets can match a proper part.

A finite set cannot match a proper part.
Therefore, it is proof that
some sets are not finite.

> No mathematics based upon two-valued logic possible.

No mathematics based upon rejecting claims for
being counter-intuitive is possible.

----
If
any claim in a sequence is
possibly-false ◇⊥
then
some claim in that sequence is
first-possibly-false #1◇⊥

Therefore,
if
each claim in a sequence is
not-first-possibly-false ¬#1◇⊥
(as q is ¬#1◇⊥
where it's preceded by p and p⇒q)
then
each claim in that sequence is
not-possibly-false ¬◇⊥

Even if
a claim is counter-intuitive, or
it refers to one of infinitely-many, or
it is not knowable which one it refers to,
such a claim is not-possibly-false ¬◇⊥

Jim Burns schrieb am Sonntag, 29. Januar 2023 um 21:08:28 UTC+1:
> On 1/29/2023 1:29 PM, WM wrote:
> > Jim Burns schrieb am Sonntag,
> > 29. Januar 2023 um 00:30:35 UTC+1:
> >> On 1/28/2023 12:44 PM, WM wrote:
>
> >>> you cannot keep Bob and you cannot delete Bob.
> >>
> >> Yes,
> >> you cannot keep Bob and you cannot delete Bob.
> >> That is closely related to the rationals
> >> in one open interval (a,b) being matchable to
> >> the rationals in any other open interval (c,d)
> >> of rationals.
> >
> > That would be the proof that "at infinity"
> > everything may happen.
> No,
> it is a proof that
> some sets can match a proper part.

If Bob cannot remain and cannot disappear either, then everything can happen without happening. Then every digit deciding about Cantor's diagonal number can be not present but not absent, i.e., not working at all. In short, your claim is nonsense.

Regards, WM

On Monday, 30 January 2023 at 00:12:55 UTC+2, WM wrote:
> Jim Burns schrieb am Sonntag, 29. Januar 2023 um 21:08:28 UTC+1:
> > On 1/29/2023 1:29 PM, WM wrote:
> > > Jim Burns schrieb am Sonntag,
> > > 29. Januar 2023 um 00:30:35 UTC+1:
> > >> On 1/28/2023 12:44 PM, WM wrote:
> >
> > >>> you cannot keep Bob and you cannot delete Bob.
> > >>
> > >> Yes,
> > >> you cannot keep Bob and you cannot delete Bob.
> > >> That is closely related to the rationals
> > >> in one open interval (a,b) being matchable to
> > >> the rationals in any other open interval (c,d)
> > >> of rationals.
> > >
> > > That would be the proof that "at infinity"
> > > everything may happen.
> > No,
> > it is a proof that
> > some sets can match a proper part.
> If Bob cannot remain and cannot disappear either, then everything can happen without happening.

If you don't have a definition of number, then what happens is ridiculously long threads such as this one which lead nowhere.

The concept of number is at the core of your arguments and yet you have no definition for it?

> Then every digit deciding about Cantor's diagonal number can be not present but not absent, i.e., not working at all. In short, your claim is nonsense.
>
> Regards, WM

Eram semper recta schrieb am Montag, 30. Januar 2023 um 09:39:58 UTC+1:

> The concept of number is at the core of your arguments and yet you have no definition for it?

There are some definitions. The basic and most secure method to establish the reality of a number is to form its fundamental set. The roman numerals are reminiscent of this method. While "2" is not a number but only a name, "II" is both, a name and a part of that number, namely of the fundamental set
2 = {all pairs like: II, you & me, mum & dad, sun & moon}.
2 is that property which all its subsets have in common. Of course this realization of
2 presupposes some a priori knowledge about 2. But here we are concerned with the
mere realization. The same method can be applied in case of 3
3 = {all triples like: III, sun & moon & earth, father & son & holy spirit}.
Although IIII is easily intelligible, and pigeons are able to distinguish up to IIIIIII
slots at first glance, it is impractical to realize larger numbers in this way. The
Romans ceased at IV already. And we would have great difficulties to identify
numbers beyond 10 in v. Neumann's rather complex construction.
Position systems, decimal or binary or other n-adic systems, have the advantage to
accomplish both identify a number and put it in order with other numbers by
economical consumption of symbols.
Does the fundamental set of 4711 exist? We don't know. It did at least in Cologne at
the beginning of the 19th century. A set of 10^1000 elements does definitely not exist as
will be shown below. Nevertheless 4711 and 10^1000 are natural numbers. Their values,
i.e., their ratios with respect to the unit are exactly determined.
It is impossible, however, to satisfy this condition for all natural numbers. It would
require an unlimited amount of resources. But the universe is finite - at least that part
available to us. Here is a simple means to realise the implications: First find out how
many different natural numbers can be stored on a 10 GB hard disk. Then, step by
step, expand the horizon to the 1011 neurons of your brain, to the 10^28 atoms of your
body, to the 10^50 atoms of our earth, to the 10^68 atoms of our galaxy, and finally to
the 10^78 protons within the universe. In principle, the whole universe could be turned
into a big computer, but with far fewer resources than is usually expected without a
thought be given to it.

Regards, WM

WM submitted this idea :
> Eram semper recta schrieb am Montag, 30. Januar 2023 um 09:39:58 UTC+1:
>
>> The concept of number is at the core of your arguments and yet you have no
>> definition for it?
>
> There are some definitions. The basic and most secure method to establish the
> reality of a number is to form its fundamental set. The roman numerals are
> reminiscent of this method. While "2" is not a number but only a name, "II"
> is both, a name and a part of that number, namely of the fundamental set 2 =
> {all pairs like: II, you & me, mum & dad, sun & moon}. 2 is that property
> which all its subsets have in common. Of course this realization of 2
> presupposes some a priori knowledge about 2. But here we are concerned with
> the mere realization. The same method can be applied in case of 3 3 = {all
> triples like: III, sun & moon & earth, father & son & holy spirit}. Although
> IIII is easily intelligible, and pigeons are able to distinguish up to
> IIIIIII slots at first glance, it is impractical to realize larger numbers in
> this way.

Wow, it's like they can count to seven.

> The Romans ceased at IV already. And we would have great
> difficulties to identify numbers beyond 10 in v. Neumann's rather complex
> construction.

Why so difficult for you when even a pigeon can do seven?

On 1/29/2023 5:12 PM, WM wrote:
> Jim Burns schrieb am Sonntag,
> 29. Januar 2023 um 21:08:28 UTC+1:
>> On 1/29/2023 1:29 PM, WM wrote:
>>> Jim Burns schrieb am Sonntag,
>>> 29. Januar 2023 um 00:30:35 UTC+1:
>>>> On 1/28/2023 12:44 PM, WM wrote:

>>>>> you cannot keep Bob and you cannot delete Bob.
>>>>
>>>> Yes,
>>>> you cannot keep Bob and you cannot delete Bob.
>>>> That is closely related to the rationals
>>>> in one open interval (a,b) being matchable to
>>>> the rationals in any other open interval (c,d)
>>>> of rationals.
>>>
>>> That would be the proof that "at infinity"
>>> everything may happen.
>>
>> No,
>> it is a proof that
>> some sets can match a proper part.
>
> If Bob cannot remain and cannot disappear either,
> then everything can happen without happening.
> Then every digit deciding about Cantor's
> diagonal number can be not present but not absent,
> i.e., not working at all.
> In short, your claim is nonsense.

No,
the assumption that Bob is always conserved
is incorrect.

The Bob we refer to here is NOT
a short, round, yellow, weird-talking
animated character who accidentally became
rightwise king born of all England.
This Bob is a particular instance of 'O'
which is moved in particular ways.

Is Bob O conserved?
Yes, if one moves Bob O some ways.
No, if one moves Bob O other ways.

As an aside,
we could ask the same question about King Bob.
How many copies of the movie "Minions" are there?
Is "Minions" conserved?
No.
And you don't think that's nonsense.

Within the "Minions" universe,
is King portrayed as Bob existing physically?
Is King Bob conserved in that portrayal?
Yes.
Because we know how physical things behave,
we know King Bob is portrayed as being conserved.

Bob O is rightwise born king of 1/2

If
Bob O is swapped in a sequence
⟨ 1/2:2/1 2/1:3/1 3/1:6/1 ... nₘ₋₁/1:nₘ/1 ⟩
in which, for each split,
some i/j:k/1 is last-before
and k/1:k₂/1 is first-after,
k = (i+j-1)(i+j-2)/2+i
k₂ = (k+1-1)(k+1-2)/2+k

then
that instance of 'O' is portrayed as
something which is conserved.

_All the swaps_ of Bob O is not
a sequence like that.

_All the swaps_ of Bob O is a sequence which
has almost all the features of
a sequence like that.

However,
not all features.
_All the swaps_ of Bob O doesn't end
_All the swaps_ of Bob O
doesn't leave Bob O anywhere
doesn't conserve Bob O

On 1/23/2023 1:16 PM, Chris M. Thomasson wrote:
> On 1/23/2023 8:21 AM, Sergi o wrote:
>> On 1/22/2023 11:24 PM, Chris M. Thomasson wrote:
>>> On 1/22/2023 7:32 PM, Sergi o wrote:
>>>> On 1/22/2023 6:02 PM, Timothy Golden wrote:
>>>>> On Sunday, January 22, 2023 at 6:57:02 PM UTC-5, Sergi o wrote:
>>>>>> On 1/22/2023 7:50 AM, WM wrote:
>>>>>>> Jim Burns schrieb am Samstag, 21. Januar 2023 um 22:20:29 UTC+1:
>>>>>>>
>>>>>>>> Therefore,
>>>>>>>> ω omega is first-after the split
>>>>>>>> {all-last-befored} {not-all-last-befored}
>>>>>>>> and nothing exists, visible or dark,
>>>>>>>> which is last-before that split.
>>>>>>>
>>>>>>> Something exists immediately before zero such that nothing fits
>>>>>>> in between. What is it?
>>>>>>>
>>>>>>> Regards, WM
>>>>>> Assume that number is X
>>>>>> however, X/2 is even closer to 0
>>>>>> therefore our assumption was wrong.
>>>>>>
>>>>>> there is no number "next to" 0
>>>>>>
>>>>>>
>>>>>> you asked this a year ago, you are still falling further behind
>>>>>>
>>>>>>
>>>>>> Prove me wrong, go ahead and try, or attempt to try, or just think
>>>>>> about it, for a min or two, bet you cannot.
>>>>>
>>>>> Already did it dick-wad.
>>>>>
>>>>> No idea what a dick-wad actually is...
>>>>>
>>>>
>>>> however, (dick-wad)/2 is closer to 0
>>>
>>> We need to define how close we need to get to zero to accomplish the
>>> task. Afaict,
>>>
>>> i[0] = 1
>>> i[1] = .1
>>> i[2] = .01
>>> i[3] = .001
>>> ...
>>>
>>> Taken to infinity gets god damn close to 0, heck it might even equal
>>> zero... ;^)
>>>
>>
>> i[13] in cm, is the diameter of Hydrogen nucleus
>>
>> i[16] in cm is Quarks
>>
>> i[33] in cm is planck length
>>
>> nutrinos have no known size
>>
>> and Heisenberg's uncertainty principle kicks in about planck length,
>> so smaller than that is a mess
>
> :^)

A mutation of a fractal even if .00000000000000000001 off mark, can
cause a MAJOR ERROR.

FromTheRafters schrieb am Montag, 30. Januar 2023 um 18:52:11 UTC+1:
> WM submitted this idea :
> > Eram semper recta schrieb am Montag, 30. Januar 2023 um 09:39:58 UTC+1:
> >
> >> The concept of number is at the core of your arguments and yet you have no
> >> definition for it?
> >
> > There are some definitions. The basic and most secure method to establish the
> > reality of a number is to form its fundamental set. The roman numerals are
> > reminiscent of this method. While "2" is not a number but only a name, "II"
> > is both, a name and a part of that number, namely of the fundamental set 2 =
> > {all pairs like: II, you & me, mum & dad, sun & moon}. 2 is that property
> > which all its subsets have in common. Of course this realization of 2
> > presupposes some a priori knowledge about 2. But here we are concerned with
> > the mere realization. The same method can be applied in case of 3 3 = {all
> > triples like: III, sun & moon & earth, father & son & holy spirit}. Although
> > IIII is easily intelligible, and pigeons are able to distinguish up to
> > IIIIIII slots at first glance, it is impractical to realize larger numbers in
> > this way.
> Wow, it's like they can count to seven.

I read that once upon a time.

Anyhow they know the much more important pigeon-hole principle: If each of the first n positive integers has a unary representation in form of a string, like ooooo, that is shorter than n then, by the pigeonhole principle, there must be two different positive integers defined by the same unary representation. Clearly this is absurd. Same holds in case of ℵo finite strings.

> > The Romans ceased at IV already. And we would have great
> > difficulties to identify numbers beyond 10 in v. Neumann's rather complex
> > construction.
> Why so difficult for you when even a pigeon can do seven?

Not in v. Neumann's language.

Regards, WM

WM expressed precisely :
> FromTheRafters schrieb am Montag, 30. Januar 2023 um 18:52:11 UTC+1:
>> WM submitted this idea :
>>> Eram semper recta schrieb am Montag, 30. Januar 2023 um 09:39:58 UTC+1:
>>>
>>>> The concept of number is at the core of your arguments and yet you have no
>>>> definition for it?
>>>
>>> There are some definitions. The basic and most secure method to establish
>>> the reality of a number is to form its fundamental set. The roman numerals
>>> are reminiscent of this method. While "2" is not a number but only a name,
>>> "II" is both, a name and a part of that number, namely of the fundamental
>>> set 2 = {all pairs like: II, you & me, mum & dad, sun & moon}. 2 is that
>>> property which all its subsets have in common. Of course this realization
>>> of 2 presupposes some a priori knowledge about 2. But here we are
>>> concerned with the mere realization. The same method can be applied in
>>> case of 3 3 = {all triples like: III, sun & moon & earth, father & son &
>>> holy spirit}. Although IIII is easily intelligible, and pigeons are able
>>> to distinguish up to IIIIIII slots at first glance, it is impractical to
>>> realize larger numbers in this way.
>> Wow, it's like they can count to seven.
>
> I read that once upon a time.
>
> Anyhow they know the much more important pigeon-hole principle: If each of
> the first n positive integers has a unary representation in form of a string,
> like ooooo, that is shorter than n then, by the pigeonhole principle, there
> must be two different positive integers defined by the same unary
> representation. Clearly this is absurd. Same holds in case of ℵo finite
> strings.
>
>>> The Romans ceased at IV already. And we would have great
>>> difficulties to identify numbers beyond 10 in v. Neumann's rather complex
>>> construction.
>> Why so difficult for you when even a pigeon can do seven?
>
> Not in v. Neumann's language.

There's an obvious unary representation there, as simple as the pigeons
had.

On Monday, 30 January 2023 at 18:09:57 UTC+2, WM wrote:
> Eram semper recta schrieb am Montag, 30. Januar 2023 um 09:39:58 UTC+1:
>
> > The concept of number is at the core of your arguments and yet you have no definition for it?
> There are some definitions. The basic and most secure method to establish the reality of a number is to form its fundamental set.

"Fundamental set"?? For crying out aloud, you believe in set theory now?

> The roman numerals are reminiscent of this method. While "2" is not a number but only a name, "II" is both, a name and a part of that number, namely of the fundamental set

So, fundamental set is defined how generically?

> 2 = {all pairs like: II, you & me, mum & dad, sun & moon}.

Or 2 = any two objects.

Only problem is that the above is circular because it contains "two".

> 2 is that property which all its subsets have in common. Of course this realization of
> 2 presupposes some a priori knowledge about 2.

Chuckle. It doesn't just presuppose; it assumes the very thing! Hence it is circular.

> But here we are concerned with the
> mere realization. The same method can be applied in case of 3
> 3 = {all triples like: III, sun & moon & earth, father & son & holy spirit}.

Suffers from the same problem as the previous example.

And so far, you haven't even touched numbers that are not multiples of the abstract unit.

> Although IIII is easily intelligible, and pigeons are able to distinguish up to IIIIIII
> slots at first glance, it is impractical to realize larger numbers in this way. The
> Romans ceased at IV already. And we would have great difficulties to identify
> numbers beyond 10 in v. Neumann's rather complex construction.
> Position systems, decimal or binary or other n-adic systems, have the advantage to
> accomplish both identify a number and put it in order with other numbers by
> economical consumption of symbols.
> Does the fundamental set of 4711 exist? We don't know. It did at least in Cologne at
> the beginning of the 19th century. A set of 10^1000 elements does definitely not exist as
> will be shown below. Nevertheless 4711 and 10^1000 are natural numbers. Their values,
> i.e., their ratios with respect to the unit are exactly determined.
> It is impossible, however, to satisfy this condition for all natural numbers. It would
> require an unlimited amount of resources. But the universe is finite - at least that part
> available to us. Here is a simple means to realise the implications: First find out how
> many different natural numbers can be stored on a 10 GB hard disk. Then, step by
> step, expand the horizon to the 1011 neurons of your brain, to the 10^28 atoms of your
> body, to the 10^50 atoms of our earth, to the 10^68 atoms of our galaxy, and finally to
> the 10^78 protons within the universe.

That's just nonsense - all of it.

> In principle, the whole universe could be turned into a big computer, but with far fewer resources than is usually expected without a
> thought be given to it.

Even if this were possible, I fail to see any relevance.

So, you've given me no definition for number, only a long rant (mostly irrelevant) with some presumptions about number already in its definition.

I give you a definition in one sentence:

A NUMBER is a NAME given to MEASURE that describes a RATIO of MAGNITUDES (not numbers!).

https://www.academia.edu/96036526/Definition_of_NUMBER_a_presentation_for_absolute_morons

>
> Regards, WM

Eram semper recta schrieb am Dienstag, 31. Januar 2023 um 19:28:09 UTC+1:
> On Monday, 30 January 2023 at 18:09:57 UTC+2, WM wrote:
> > Eram semper recta schrieb am Montag, 30. Januar 2023 um 09:39:58 UTC+1:
> >
> > > The concept of number is at the core of your arguments and yet you have no definition for it?
> > There are some definitions. The basic and most secure method to establish the reality of a number is to form its fundamental set.
> "Fundamental set"?? For crying out aloud, you believe in set theory now?

Finite set theory is unquestinable.

> > The roman numerals are reminiscent of this method. While "2" is not a number but only a name, "II" is both, a name and a part of that number, namely of the fundamental set
> So, fundamental set is defined how generically?
> > 2 = {all pairs like: II, you & me, mum & dad, sun & moon}.
> Or 2 = any two objects.
>
> Only problem is that the above is circular because it contains "two".

Everything is circular before you start without any word.

> > 2 is that property which all its subsets have in common. Of course this realization of
> > 2 presupposes some a priori knowledge about 2.
> Chuckle. It doesn't just presuppose; it assumes the very thing! Hence it is circular.

Explain something to someone who does not know any word.

> A NUMBER is a NAME given to MEASURE that describes a RATIO of MAGNITUDES (not numbers!).

Circular, like every explanation.

Regards, WM