Ben Bacarisse schrieb am Dienstag, 7. Februar 2023 um 01:57:20 UTC+1:
> Fritz Feldhase <franz.fri...@gmail.com> writes:

> > The reason -it seems- is that Mr Mückenheim is suffering from
> > quantifier dyslexia.
> Yes, that's a common one.

Can you understand that the unit fractions stretch immediately to zero?
Common quantifier magic claims:
For all FISONs there are ℵo natnumbers as successors, but the set of all FISONs has no natnumber as successors.
That nonsense has now been refuted.

> Common too is the belief that what is it true
> of all elements should be true of the set of elements.

If all fractions shall be enumerated, then none must be missing. That cannot be accomplished by quantifier magic. That must be done until the last one.. But that is impossible.

Regards, WM

On Tuesday, February 7, 2023 at 12:35:24 PM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 7. Februar 2023 um 11:08:06 UTC+1:
>
> > Hint: For each and every natural number n: card {1/m : m e IN, m > n} = ℵo.
> >
> > This means: There is no natural number n such that card {1/m : m e IN, m > n} =/= ℵo.
> >
> There is no definable natural number with less successors, and

I said: There is _no natural number_ n such that card {1/m : m e IN, m > n} =/= ℵo. Period.

> dark numbers [...] have no discernible order.

This proves that they aren't natural numbers.

Hint: The set of natual numbers is a totaly ordered set (where the order is the usual order on IN).

See: https://en.wikipedia.org/wiki/Total_order

Learn some logic, learn some math, Mückenheim.

On Tuesday, February 7, 2023 at 12:49:55 PM UTC+1, WM wrote:

> For all [elements in all] FISONs there are ℵo natnumbers as successors, but the set of all FISONs has no natnumber as successors.

Why should it?! After all _the set of all FISONs_ isn't itself a natural number. Hence there's no need for it to have a natural number as successor.

Ben: "Common too is the belief that what is it true of all elements [of a set] should be true of the set of elements [too]."

Indeed!

Fritz Feldhase schrieb am Dienstag, 7. Februar 2023 um 12:54:27 UTC+1:
> On Tuesday, February 7, 2023 at 12:35:24 PM UTC+1, WM wrote:

> > There is no definable natural number with less successors, and
> I said: There is _no natural number_ n such that card {1/m : m e IN, m > n} =/= ℵo. Period.
>
I know, but that is wrong. In (0, 1] there are unit fractions without successors.

> > dark numbers [...] have no discernible order.
>
> This proves that they aren't natural numbers.

No, that is wrong too if Cantor is right: ω - n = ω.
>
> Hint: The set of natual numbers is a totaly ordered set (where the order is the usual order on IN).

> Learn some logic, learn some math, Mückenheim.

Fortunately I know just enoughof both to know that your "logic" is nonsense..

"Mathematical logic" has completely deformed the thinking of mathematicians and of philosophers, [V.48] [L. Wittgenstein: "Remarks on the foundations of mathematics", Wiley-Blackwell (1991)]

According to this {{Brouwer's}} view and reading of history, classical logic was abstracted from the mathematics of finite sets and their subsets. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. [H. Weyl: "Levels of infinity: Selected writings on mathematics and philosophy", Peter Pesic (ed.), Dover Publications (2012) p. 140f]

Have you not yet understood this self-evident truth?

Almost all natural numbers are following upon every natural number?
The set contains more than every such natnumber and is followed by no natnumber?
Only an ausgesprochener Schwachkopf kann das behaupten.

Regards, WM

WM has brought this to us :
> William schrieb am Montag, 6. Februar 2023 um 23:29:59 UTC+1:
>> On Monday, February 6, 2023 at 6:22:09 PM UTC-4, WM wrote:
>>> William schrieb am Montag, 6. Februar 2023 um 20:05:22 UTC+1:
>>>> On Monday, February 6, 2023 at 2:23:43 PM UTC-4, WM wrote:
>>>>> William schrieb am Montag, 6. Februar 2023 um 17:22:39 UTC+1:
>>>>>> the things you call "dark natural numbers" are not natural numbers.
>>>>> They are natural numbers
>>>> Nope, the reciprocal of a natural number is a real number. The reciprocal
>>>> or the things you call "dark natural numbers" are smaller than any element
>>>> of U_P
>>> Not all! Many can be made to belong to U_P, if you try hard enough.
>> Nope, no matter how hard you try, the things you call "dark natural numbers"
>> are larger than any element of |N_P
>
> Yes, most of them are.

Almost all of them are. The thing is, they're not dark.

WM explained :
> Fritz Feldhase schrieb am Dienstag, 7. Februar 2023 um 11:30:51 UTC+1:
>> On Tuesday, February 7, 2023 at 11:10:25 AM UTC+1, WM wrote:
>
>>> But their reciprocals lie as points on the positive real line.
>> No, the don't. They are infinitesimal numbers. There are no such numbers in
>> IR (and hence not on the "positive real line").
>
> All unit fractions are points in the interval (0, 1].

Yes, but only the rational points therein are addressed by unit
fractions. There are always irrational points between zero and a
nonzero rational number too, don't forget them.

WM pretended :
> Fritz Feldhase schrieb am Dienstag, 7. Februar 2023 um 11:08:06 UTC+1:
>
>> Hint: For each and every natural number n: card {1/m : m e IN, m > n} = ℵo.
>>
>> This means: There is no natural number n such that card {1/m : m e IN, m >
>> n} =/= ℵo.
>
> There is no definable natural number with less successors,

Sure, in the normal ordering each successor is greater than its
predecessor.

WM used his keyboard to write :
> Ben Bacarisse schrieb am Montag, 6. Februar 2023 um 21:28:31 UTC+1:
>> WM <askas...@gmail.com> writes:
>
>>> That means there is no unit fraction between all unit fractions and
>>> zero.
>> ~∃n : ∀m : 0 < 1/n < 1/m
>>> But there are ℵo unit fractions between every chosen unit fraction and
>>> zero.
>> ∀n : |{1/m | 0 < 1/m < 1/n}| = ℵ₀
>>
>> or (more simply)
>>
>> ∀n : ∃m : 0 < 1/m < 1/n
>>
>> as one can derive the fact that {1/m | 0 < 1/m < 1/n} is infinite (for
>> any 1/n) from this simpler fact.
>>> Can you understand this when you are trying very, very hard?
>> For some reason you think it is worth saying these facts again and again
>> as if they were ever in dispute,
>
> In dispute was the fact that the unit fractions (not only the set of unit
> fractions or the interval of unit fractions) stretch immediately to zero.

Explain what 'stretch immediately' means in mathematics.

FromTheRafters schrieb am Dienstag, 7. Februar 2023 um 14:43:30 UTC+1:
> WM explained :
> > Fritz Feldhase schrieb am Dienstag, 7. Februar 2023 um 11:30:51 UTC+1:
> >> On Tuesday, February 7, 2023 at 11:10:25 AM UTC+1, WM wrote:
> >
> >>> But their reciprocals lie as points on the positive real line.
> >> No, the don't. They are infinitesimal numbers. There are no such numbers in
> >> IR (and hence not on the "positive real line").
> >
> > All unit fractions are points in the interval (0, 1].
> Yes, but only the rational points therein are addressed by unit
> fractions. There are always irrational points between zero and a
> nonzero rational number too, don't forget them.

The unit fractions alone already stretch immediately to zero. That means there is nothing in between.

Regards, WM

On Tuesday, February 7, 2023 at 6:10:25 AM UTC-4, WM wrote:
> William schrieb am Montag, 6. Februar 2023 um 23:29:59 UTC+1:
> >... no matter how hard you try, the things you call "dark natural numbers" are larger than any element of |N_P
> Yes, most of them are.
Nope, each. There are no exceptions,

> > so their reciprocals are smaller than any element of U_P
> But their reciprocals lie as points on the positive real line.
Nope, nothing that is smaller than any element of U_P is an element of |R. The real line only has points that correspond to elements of |R. The things you call "dark natural numbers" are not natural numbers.

William schrieb am Dienstag, 7. Februar 2023 um 16:16:32 UTC+1:
> On Tuesday, February 7, 2023 at 6:10:25 AM UTC-4, WM wrote:
> > William schrieb am Montag, 6. Februar 2023 um 23:29:59 UTC+1:
> > >... no matter how hard you try, the things you call "dark natural numbers" are larger than any element of |N_P
> > Yes, most of them are.
> Nope, each. There are no exceptions,

Of course. Simply name the largest known natnumber. I will name a larger one. The border between visible and dark is permeable. Only the border between dark natnumbers and omega is sharp like between 0 and (0, 1].

> > > so their reciprocals are smaller than any element of U_P
> > But their reciprocals lie as points on the positive real line.
> Nope, nothing that is smaller than any element of U_P is an element of |R.. The real line only has points that correspond to elements of |R.

Then find the first positive point. Fail.

> The things you call "dark natural numbers" are not natural numbers.

The points closest to zero that you can find are not really the points closest to zero. But of course there are points closest to zero because the geometry does not depend on the inductive explanation "if there is a point then there is a point closer to zero".

They are all there already, whether or not you have named one or the other!

Regards, WM

On Tuesday, February 7, 2023 at 11:36:55 AM UTC-4, WM wrote:
> William schrieb am Dienstag, 7. Februar 2023 um 16:16:32 UTC+1:
> > On Tuesday, February 7, 2023 at 6:10:25 AM UTC-4, WM wrote:
> > > William schrieb am Montag, 6. Februar 2023 um 23:29:59 UTC+1:
> > > >... no matter how hard you try, the things you call "dark natural numbers" are larger than any element of |N_P
> > > Yes, most of them are.
> > Nope, each. There are no exceptions,
> Of course. Simply name the largest known natnumber. I will name a larger one.

You will name an element of |N_P, not a "dark natural number". No element of |N_P is a "dark natural number".

On Tuesday, February 7, 2023 at 10:36:55 AM UTC-5, WM wrote:
> William schrieb am Dienstag, 7. Februar 2023 um 16:16:32 UTC+1:
> > On Tuesday, February 7, 2023 at 6:10:25 AM UTC-4, WM wrote:
> > > William schrieb am Montag, 6. Februar 2023 um 23:29:59 UTC+1:
> > > >... no matter how hard you try, the things you call "dark natural numbers" are larger than any element of |N_P
> > > Yes, most of them are.
> > Nope, each. There are no exceptions,
> Of course. Simply name the largest known natnumber. I will name a larger one. The border between visible and dark is permeable. Only the border between dark natnumbers and omega is sharp like between 0 and (0, 1].
> > > > so their reciprocals are smaller than any element of U_P
> > > But their reciprocals lie as points on the positive real line.
> > Nope, nothing that is smaller than any element of U_P is an element of |R. The real line only has points that correspond to elements of |R.
> Then find the first positive point. Fail.

The first positive point is:
0.000...01
This is of course an inductively defined value.
The usage of the decimal point will have to be investigated in order to gain nonrational support. The usage of the ellipses, much as you'd like to deny them, really will have to be let in since you've already put them to use so many times. The freedom to construct such a value deserves to be taken, and it has been taken. There is a halting problem in the construction but so it is in all of this analysis.

Beyond these details, Peano seems to be a radix one construction, and under those circumstances the meaning of the decimal point is dubious. Even the usage of the zeros is dubious. But then, so was your usage of the value '3', and '2'. The very concept of number as an encrypted form is put on the table here. We exist in a system which is badly accumulated. There were no compilers; no data structures; when it evolved. Somewhere along the way the operators and the values got mixed, and they don't really mix very well. The rational value that this supposed analysis sits on is one of these things. There is nothing fundamental about it. Evaluate it and we'll be into that decimal point. But you see, that decimal point can and should stand freely of the conflicted construction that is the rational value. So we can arrive at a secondary unital concept, and the rationals can actually be reconstructed on this same concept, as when thirds are achieved by treating the value three as one, but the radix form is so close by that the misattribution of the rational numbers is best felt while investigating the radix numbers, which arguably came first.

> > The things you call "dark natural numbers" are not natural numbers.
> The points closest to zero that you can find are not really the points closest to zero. But of course there are points closest to zero because the geometry does not depend on the inductive explanation "if there is a point then there is a point closer to zero".
>
> They are all there already, whether or not you have named one or the other!
>
> Regards, WM

On Tuesday, February 7, 2023 at 4:36:55 PM UTC+1, WM wrote:

> there are points [=/= 0] closest to zero

No, there aren't.

If x is a point =/= 0 then x/2 is closer to zero.

Hint: Ax e IR\{0} : 0 < |x/2| < |x|.

What's the matter with you, Mückenheim? Did you brain fall out?

On Tuesday, February 7, 2023 at 1:44:51 PM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 7. Februar 2023 um 12:54:27 UTC+1:
> > On Tuesday, February 7, 2023 at 12:35:24 PM UTC+1, WM wrote:
> >
> > I said: There is _no natural number_ n such that card {1/m : m e IN, m > n} =/= ℵo. Period.
> >
> I know, but that is wrong.

No. It isn't.

> In (0, 1] there are unit fractions without successors.

Nope.

If x is a unit fraction in (0, 1] then [by definition] there's a natural number n such that x = 1/n.
Then /(n+1) - i.e the successor of 1/n = x - is a unit fraction which is in (0, 1] too.

Hint: There are no unit fractions without successors in (0, 1] because there are no natural numbers without successors in IN, you know.

Remember that there's a Peano axiom: An e IN: s(n) e IN.

What's the matter with you, Mückenheim? Did you brain fall out?

on 2/7/2023, WM supposed :
> FromTheRafters schrieb am Dienstag, 7. Februar 2023 um 14:43:30 UTC+1:
>> WM explained :
>>> Fritz Feldhase schrieb am Dienstag, 7. Februar 2023 um 11:30:51 UTC+1:
>>>> On Tuesday, February 7, 2023 at 11:10:25 AM UTC+1, WM wrote:
>>>>> But their reciprocals lie as points on the positive real line.
>>>> No, the don't. They are infinitesimal numbers. There are no such numbers
>>>> in IR (and hence not on the "positive real line").
>>>
>>> All unit fractions are points in the interval (0, 1].
>> Yes, but only the rational points therein are addressed by unit
>> fractions. There are always irrational points between zero and a
>> nonzero rational number too, don't forget them.
>
> The unit fractions alone already stretch immediately to zero. That means
> there is nothing in between.

How can they when zero isn't even there?

On Tuesday, February 7, 2023 at 1:44:51 PM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 7. Februar 2023 um 12:54:27 UTC+1:
> >
> > Learn some logic, learn some math, Mückenheim.
> >
> Fortunately I know just enough of both to know <bla>

Alternatiely you may read https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.64.2655

On Tuesday, February 7, 2023 at 1:44:51 PM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 7. Februar 2023 um 12:54:27 UTC+1:
> >
> > Learn some logic, learn some math, Mückenheim.
> >
> Fortunately I know just enough of both to know <bla>

Alternatively you may read this: https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.64.2655

On Tuesday, February 7, 2023 at 1:44:51 PM UTC+1, WM wrote:

> Almost all natural numbers are following upon every natural number?

Indeed!

An e IN: {k e IN : k > n} = IN \ {1, ..., n} & {1, ..., n} is finite.

On 2/7/2023 3:03 AM, WM wrote:
> William schrieb am Montag, 6. Februar 2023 um 23:29:59 UTC+1:
>> On Monday, February 6, 2023 at 6:22:09 PM UTC-4, WM wrote:
>>> William schrieb am Montag, 6. Februar 2023 um 20:05:22 UTC+1:
>>>> On Monday, February 6, 2023 at 2:23:43 PM UTC-4, WM wrote:
>>>>> William schrieb am Montag, 6. Februar 2023 um 17:22:39 UTC+1:
>>>>
>>>>>> the things you call "dark natural numbers" are not natural numbers.
>>>>> They are natural numbers
>>>> Nope, the reciprocal of a natural number is a real number. The reciprocal or the things you call "dark natural numbers" are smaller than any element of U_P
>>> Not all! Many can be made to belong to U_P, if you try hard enough.
>> Nope, no matter how hard you try, the things you call "dark natural numbers" are larger than any element of |N_P so their reciprocals are smaller than any element of U_P
>
> Not quite. Many can be made visible. But most cannot.

Name a natural number that cannot be made visible?

[...]

On 2/7/2023 5:47 AM, FromTheRafters wrote:
> WM pretended :
>> Fritz Feldhase schrieb am Dienstag, 7. Februar 2023 um 11:08:06 UTC+1:
>>
>>> Hint: For each and every natural number n: card {1/m : m e IN, m > n}
>>> = ℵo.
>>> This means: There is no natural number n such that card {1/m : m e
>>> IN, m > n} =/= ℵo.
>>
>> There is no definable natural number with less successors,
>
> Sure, in the normal ordering each successor is greater than its
> predecessor.

WM must be trolling?

On 2/7/2023 11:08 AM, Timothy Golden wrote:
> On Tuesday, February 7, 2023
> at 10:36:55 AM UTC-5, WM wrote:

>> Then find the first positive point. Fail.
>
> The first positive point is:
> 0.000...01

The line in which, for any point,
a first-after or a last-before point exists
is not the correct line.

| The Tao that can be spoken of is not
| the enduring and unchanging Tao.
| The name that can be named is not
| the enduring and unchanging name.
| -- Tao te Ching

>> But of course there are points closest to zero
>> because the geometry does not depend on the
>> inductive explanation "if there is a point
>> then there is a point closer to zero".

The geometry has, for each line segment.
two intersecting circles of the same radius
each centered at one end point of the segment.

The geometry has a line segment from one
point of intersection to the other, which
intersects the first segment at its midpoint.

Thus,
the geometry has, for any twp points,
the midpoint between those two points.

>> But of course there are points closest to zero
>> because the geometry does not depend on the
>> inductive explanation "if there is a point
>> then there is a point closer to zero".

If there is zero and a non-zero point
then there is a point closer to zero.

On 2/7/2023 2:35 PM, Jim Burns wrote:
> On 2/7/2023 11:08 AM, Timothy Golden wrote:
>> On Tuesday, February 7, 2023
>> at 10:36:55 AM UTC-5, WM wrote:
>
>>> Then find the first positive point. Fail.
>>
>> The first positive point is:
>> 0.000...01
>
> The line in which, for any point,
> a first-after or a last-before point exists
> is not the correct line.
>
> | The Tao that can be spoken of is not
> | the enduring and unchanging Tao.
> | The name that can be named is not
> | the enduring and unchanging name.
> |
> -- Tao te Ching
>
>
>>> But of course there are points closest to zero
>>> because the geometry does not depend on the
>>> inductive explanation "if there is a point
>>> then there is a point closer to zero".
>
> The geometry has, for each line segment.
> two intersecting circles of the same radius
> each centered at one end point of the segment.
>
> The geometry has a line segment from one
> point of intersection to the other, which
> intersects the first segment at its midpoint.
>
> Thus,
> the geometry has, for any twp points,
> the midpoint between those two points.
>
>>> But of course there are points closest to zero
>>> because the geometry does not depend on the
>>> inductive explanation "if there is a point
>>> then there is a point closer to zero".
>
> If there is zero and a non-zero point
> then there is a point closer to zero.
>

Right. There is always a point that is closer to zero than any given
non-zero point.

point_0
point_1

difference = point_1 - point_0;

mid_point = point_0 + difference / 2;

There, we just got a mid point.

FromTheRafters schrieb am Dienstag, 7. Februar 2023 um 17:45:17 UTC+1:
> on 2/7/2023, WM supposed :

> > The unit fractions alone already stretch immediately to zero. That means
> > there is nothing in between.
> How can they when zero isn't even there?

Zero is there, immediately next to the points of the interval.

Regards, WM

Fritz Feldhase schrieb am Dienstag, 7. Februar 2023 um 17:28:46 UTC+1:

> > In (0, 1] there are unit fractions without successors.
> Nope.

In (0, 1] there are points immediately next to zero, i.e., without infinitely many points in between. Proof: If you cut anything of the interval, then you delete such points.

> Remember that there's a Peano axiom: An e IN: s(n) e IN.

Wrong for dark numbers. But for fractions it is obvious.

Regards, WM