A number that cannot be determined

[WM]

It is shown by the discreteness of unit fractions that not every real number can be determined.

Define the function NUF(x) measuring the Number of Unit Fractions between 0 and x. Clearly NUF(x) = 0 for x ≤ 0 and NUF(x) = ℵ0 for every x > 0 that can be determined. But not for all x!

The function NUF(x) is a step-function. It can increase from 0 at x = 0 to greater values, either in a step of size 1 or in a step of size more than 1. But increase by more than 1 is excluded by the gaps between unit fractions:

∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0.

Note the universal quantifier, according to which never (and in no limit) two unit fractions occupy the same point x. Therefore the step size can only be 1, resulting in a real x with NUF(x) = 1. This point x however, and all points where NUF(x) < ℵ0, cannot be determined.

Regards, WM

[markus...@gmail.com]

What does NUF(x) = ℵ0 mean? What's a determinable number?

[WM]

> What does NUF(x) = ℵ0 mean?

NUF(x) = ℵ0 means that the number of unit fractions between 0 and x is actually infinite, ℵ0.

What's a determinable number?

A determinable number is a number that can be communicated between a sender and a receiver such that both know the quantity described by that number. Examples: 7 or π or greatest prime number known in 2022 or height of the Mount Everest in meters.

Regards, WM

[mitchr...@gmail.com]

On Saturday, July 29, 2023 at 4:00:28 AM UTC-7, WM wrote:
> It is shown by the discreteness of unit fractions that not every real number can be determined.

Do they not exist in concept?
i is only formula concept that has no solution...

[Jim Burns]

On 7/29/2023 9:27 AM, WM wrote:
> markus...@gmail.com schrieb am Samstag,
> 29. Juli 2023 um 14:34:40 UTC+2:
>> lördag 29 juli 2023 kl. 13:00:28 UTC+2
>> skrev WM:

>>> not every real number can be determined.

>> What's a determinable number?
>
> A determinable number is
> a number that can be communicated
> between a sender and a receiver such that
> both know the quantity described by
> that number.

A determinable real number
is the least upper bound of
a bounded non-∅ subset of ℚ
and can be communicated (&c).

A NON-determinable real number
is the least upper bound of
a bounded non-∅ subset of ℚ
and canNOT be communicated (&c).

For each real number x
we might not know if x is
determinable or non-determinable,
but we know that x is
the least upper bound of
a bounded non-∅ subset of ℚ

Because
we know that the claim
| x is the least upper bound of
| a bounded non-∅ subset of ℚ
|
is true of any real number x,
we can augment that claim with
not-first-false claims, and
we will know that
the augmenting claims are also
true of any real number x.

Still without knowing if x is
determinable or non-determinable.

[bassam karzeddin]

A number that can't be determined is so simply No number
Where the numbers that can be determined are only the positive constructible numbers
BKK

[Chris M. Thomasson]

On 7/29/2023 10:47 AM, bassam karzeddin wrote:
[...]

> A number that can't be determined is so simply No number
> Where the numbers that can be determined are only the positive constructible numbers

Can sqrt(2) be constructed?

[markus...@gmail.com]

What is the domain and codomain of NUF?

I don't think your definition of a "determinable number" makes much mathematical sense. Can you formulate the definition in mathematical terms?

[Fritz Feldhase]

On Saturday, July 29, 2023 at 7:51:33 PM UTC+2, Chris M. Thomasson wrote:

> Can sqrt(2) be constructed?

Yeah, by ruler and a compass.

https://mcadamsmath.tripod.com/numbers/cons_sqrt_2.html

or:

https://youtu.be/JJ34Fq8ydd0

[Chris M. Thomasson]

On 7/29/2023 11:46 AM, Fritz Feldhase wrote:
> On Saturday, July 29, 2023 at 7:51:33 PM UTC+2, Chris M. Thomasson wrote:
>
>> Can sqrt(2) be constructed?
>
> Yeah, by ruler and a compass.

Right. Draw the unit square, the diagonal is sqrt(2). :^)

I was wondering if The King bassam karzeddin thought that sqrt(2) was
some impossible number.

[Fritz Feldhase]

On Saturday, July 29, 2023 at 8:38:50 PM UTC+2, markus...@gmail.com wrote:

> What is the domain and codomain of NUF?
>
> I don't think your definition of a "determinable number" makes much mathematical sense. Can you formulate the definition in mathematical terms?

dom(NUF) = IR
img(NUF) = {0, aleph_0}

Actually, NUF(x) := card {q e {1/n : n e IN} : q <= x} (x e IR). Hence img(NUF) = {0, aleph_0}.

"determinable number" is Mückenheim nonsense.

[Fritz Feldhase]

On Saturday, July 29, 2023 at 8:50:15 PM UTC+2, Chris M. Thomasson wrote:
> On 7/29/2023 11:46 AM, Fritz Feldhase wrote:
> > On Saturday, July 29, 2023 at 7:51:33 PM UTC+2, Chris M. Thomasson wrote:
> >
> >> Can sqrt(2) be constructed?
> >
> > Yeah, by ruler and a compass.
> Right. Draw the unit square, the diagonal is sqrt(2). :^)
>
> I was wondering if The King bassam karzeddin thought that sqrt(2) was
> some impossible number.

I c. Who knows...

[WM]

mitchr...@gmail.com schrieb am Samstag, 29. Juli 2023 um 18:58:21 UTC+2:
> On Saturday, July 29, 2023 at 4:00:28 AM UTC-7, WM wrote:
> > It is shown by the discreteness of unit fractions that not every real number can be determined.
> Do they not exist in concept?

They exit if Cantor's actual infinity is true. Otherwise they do not exist.

Regards, WM

[mitchr...@gmail.com]

And where is the proof of your math?

> Regards, WM

[WM]

mitchr...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 00:00:43 UTC+2:
> On Saturday, July 29, 2023 at 2:57:35 PM UTC-7, WM wrote:
> > mitchr...@gmail.com schrieb am Samstag, 29. Juli 2023 um 18:58:21 UTC+2:
> > > On Saturday, July 29, 2023 at 4:00:28 AM UTC-7, WM wrote:
> > > > It is shown by the discreteness of unit fractions that not every real number can be determined.
> > > Do they not exist in concept?

> > They exist if Cantor's actual infinity is true. Otherwise they do not exist.

> And where is the proof of your math?

It is this: ℵo unit fractions and their internal distances occupy an interval D larger than 0.

∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0.

NUF(0) = 0.
Therefore
∀x ∈ (0, 1]: NUF(x) = ℵo
is wrong whereas
∀x ∈ (D, 1]: NUF(x) = ℵo is true.

Regards

[WM]

markus...@gmail.com schrieb am Samstag, 29. Juli 2023 um 20:38:50 UTC+2:
> lördag 29 juli 2023 kl. 15:27:41 UTC+2 skrev WM:
> > markus...@gmail.com schrieb am Samstag, 29. Juli 2023 um 14:34:40 UTC+2:
> > > lördag 29 juli 2023 kl. 13:00:28 UTC+2 skrev WM:
> > > > It is shown by the discreteness of unit fractions that not every real number can be determined.
> > > >
> > > > Define the function NUF(x) measuring the Number of Unit Fractions between 0 and x. Clearly NUF(x) = 0 for x ≤ 0 and NUF(x) = ℵ0 for every x > 0 that can be determined. But not for all x!
> > > >
> > > > The function NUF(x) is a step-function. It can increase from 0 at x = 0 to greater values, either in a step of size 1 or in a step of size more than 1. But increase by more than 1 is excluded by the gaps between unit fractions:
> > > >
> > > > ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0.
> > > >
> > > > Note the universal quantifier, according to which never (and in no limit) two unit fractions occupy the same point x. Therefore the step size can only be 1, resulting in a real x with NUF(x) = 1. This point x however, and all points where NUF(x) < ℵ0, cannot be determined.
> > > >
> > > What does NUF(x) = ℵ0 mean?
> > NUF(x) = ℵ0 means that the number of unit fractions between 0 and x is actually infinite, ℵ0.
> > What's a determinable number?
> > A determinable number is a number that can be communicated between a sender and a receiver such that both know the quantity described by that number. Examples: 7 or π or greatest prime number known in 2022 or height of the Mount Everest in meters.
> >

> What is the domain and codomain of NUF?

Domain is ℝ, co-domain is [0, ℵo]

>
> I don't think your definition of a "determinable number" makes much mathematical sense.

Determinable numbers are the basis of classical mathematics. Determinability is so natural there that it is never mentioned.

Regards, WM

[WM]

Fritz Feldhase schrieb am Samstag, 29. Juli 2023 um 20:52:29 UTC+2:

> img(NUF) = {0, aleph_0}

Nonsense.

ℵo unit fractions and their internal distances occupy an interval D larger than 0. Therefore

[gwen w]

om- wow h-lls-t- of sci.math wow. added to list haha. i can not immediately prov textbook of forml representation but putting my pennies on WM here. brb

[bassam karzeddin]

I don't know why this site isn't providing the faciicularlity of replying a particular post, but only the facility of replying the last post only as "reply to all "

Is this applicable to every member or it is only for myself? Wonder!

I also noted that some annymous acadimic theif is writing a very ugly reply on my name without posting it, where I deleted, beside the delibratly difficulty while replying!

Is this truly a moderated site or what kind of corruption behind this unfamiliar acts?

At any case, & since replying facility to a particular post isn't available for myself, I would reply to a post or a question for myself asked by Chris M Thomasson , asking me a very foolish question that had been answered thousands of years back for trillions of times & hundreds of times by myself

The very stupid question was about construction of Sqrt2 (Immagin?)

What is going on truly in this very SH**TY & Trolish site indeed?
Who are the annymous hired acadimic Trolls 🧌 behind all that big nonsense?
And who are the unknown masters behind the scene who are hiring them & why?

Are they Donkypedia writeers or Journals publishers or Universities masters or books authors in mathematics who want desperately to protect their huge false SH**T in mathematics? No woundrs!

Every thing is possible for sure!

Bassam Karzeddin

[markus...@gmail.com]

Okay, but can you provide a MATHEMATICAL FORMAL DEFINITION for what s determinable number is?

[WM]

> Okay, but can you provide a MATHEMATICAL FORMAL DEFINITION for what s determinable number is?

In classical mathematics only determinable numbers are numbers. Therefore they are simply what "number" means in classical mathematics.

As an aside: "Formal" means a special kind of mathematics. There are many other kinds. Mathematicians who are unable to think without crutches need it "formal".

Regards, WM

[markus...@gmail.com]

What is "classical mathematics" and why are you unable to give a precise formal definition of what a "determinable number" is?

[WM]

> What is "classical mathematics"

I understand by classical mathematiss mathematics without actual infinity.

> and why are you unable to give a precise formal definition of what a "determinable number" is?

Why are you unable to give a formal definition what a set is?
These things are required before mathematics starts.

Regards, WM

[markus...@gmail.com]

I'm not unable to give a formal definition of what a set is. A set is any object that satisfies the axioms of whenever set theory you are dealing with. A ZFC-set is an object that satisfies the ZFC axioms.

A very precise and concrete definition. I can even state them with logical symbols.

Now, can you define what a "determinable number" is? All I've heard so far are excuses to why you can't. Not a good start. 🤨

Classical mathematics, as defined as the mathematics you saw in ancient Greece, did indeed have the concept of infinity.

[WM]

markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 16:11:57 UTC+2:
> söndag 30 juli 2023 kl. 16:02:18 UTC+2 skrev WM:

> I'm not unable to give a formal definition of what a set is. A set is any object that satisfies the axioms of whenever set theory you are dealing with. A ZFC-set is an object that satisfies the ZFC axioms.

What is an object that satisfies the axioms?

>
> A very precise and concrete definition.

No.

>
> Now, can you define what a "determinable number" is? All I've heard so far are excuses to why you can't. Not a good start. 🤨

A determinable number is a number that can be communicated as an individual.

>
> Classical mathematics, as defined as the mathematics you saw in ancient Greece, did indeed have the concept of infinity.

The concept of infinity from Greeks to Gauss differs from actual infinity.

Regards, WM

[Fritz Feldhase]

On Sunday, July 30, 2023 at 4:11:57 PM UTC+2, markus...@gmail.com wrote:

> /Classical mathematics/, as defined as the mathematics you saw in ancient Greece

This is not how this term (usually) is understood.

Hint:

"In the foundations of mathematics, /classical mathematics/ refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory. It stands in contrast to other types of mathematics such as constructive mathematics or predicative mathematics. In practice, the most common non-classical systems are used in constructive mathematics." (Wikipedia)

Ancient Greek math is just that: /Greek mathematics/.

"/Greek mathematics/ refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly attested from the late 7th century BC to the 6th century AD, around the shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire region, from Anatolia to Italy and North Africa, but were united by Greek culture and the Greek language.[1] The development of mathematics as a theoretical discipline and the use of proofs is an important difference between Greek mathematics and those of preceding civilizations." (Wikipedia)

[Fritz Feldhase]

On Sunday, July 30, 2023 at 3:52:46 PM UTC+2, markus...@gmail.com wrote:

> What is "classical mathematics"

"/classical mathematics/ refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory.[1] It stands in contrast to other types of mathematics such as constructive mathematics or predicative mathematics. In practice, the most common non-classical systems are used in constructive mathematics.""

Source: https://en.wikipedia.org/wiki/Classical_mathematics

[Fritz Feldhase]

On Sunday, July 30, 2023 at 4:02:18 PM UTC+2, WM wrote:

> > What is "classical mathematics"
>
> I understand by classical mathematiss mathematics without actual infinity.

The problem is that you do NOT understand ANYTHING in math, you psyhotic asshole full of shit!

Hint: "In the foundations of mathematics, classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory. It stands in contrast to other types of mathematics such as constructive mathematics or predicative mathematics. In practice, the most common non-classical systems are used in constructive mathematics.

Classical mathematics is sometimes attacked on philosophical grounds, due to constructivist and other objections to the logic, set theory, etc., chosen as its foundations, such as have been expressed by L. E. J. Brouwer. Almost all mathematics, however, is done in the classical tradition, or in ways compatible with it.

Defenders of classical mathematics, such as David Hilbert, have argued that it is easier to work in, and is most fruitful; although they acknowledge non-classical mathematics has at times led to fruitful results that classical mathematics could not (or could not so easily) attain, they argue that on the whole, it is the other way round."

Source: https://en.wikipedia.org/wiki/Classical_mathematics

So, yes, "classical mathematics" is VERY MUCH about infinite entities (called sets).

[Fritz Feldhase]

On Sunday, July 30, 2023 at 12:11:24 AM UTC+2, WM wrote:

> ℵo unit fractions and their internal distances occupy an interval D larger than 0.

This claim doen't make any sense. Especially in the cotext of the following claim:

> ∀x ∈ (D, 1]: NUF(x) = ℵo is true.

LOOK DUMBO, D *obviously* is a real number here (since it refers as an endpoint of an interval).

But if D is a real number, the statement "ℵo unit fractions and their internal distances occupy an interval D ..." is NONSENS.

On the other hand, with the phrase "...D larger than 0" you again seem to refer to a REAL NUMBER D > 0.

So you claim should read: "ℵo unit fractions and their internal distances occupy an interval ___________". << please fill in the blank.

Hint: For any real number D > 0 "ℵo unit fractions and their internal distances occupy" the intervall (0, D].

(Actually, for each and every real number D > 0 ℵo unit fractions are smaller than D (but larger than 0).)

[Dieter Heidorn]

markus...@gmail.com schrieb:

> söndag 30 juli 2023 kl. 16:02:18 UTC+2 skrev WM:
>> markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 15:52:46 UTC+2:
>>> söndag 30 juli 2023 kl. 13:20:01 UTC+2 skrev WM:
>>>> markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 09:33:11 UTC+2:
>>>>> söndag 30 juli 2023 kl. 00:21:38 UTC+2 skrev WM:
>>>>>> markus...@gmail.com schrieb am Samstag, 29. Juli 2023 um 20:38:50 UTC+2:
>>>>>>> lördag 29 juli 2023 kl. 15:27:41 UTC+2 skrev WM:
>>>>>>>> markus...@gmail.com schrieb am Samstag, 29. Juli 2023 um 14:34:40 UTC+2:
>>>>>>>>> lördag 29 juli 2023 kl. 13:00:28 UTC+2 skrev WM:

>>>>>>>> What's a determinable number?
>>>>>>>> A determinable number is a number that can be communicated between a sender and a receiver such that both know the quantity described by that number. Examples: 7 or π or greatest prime number known in 2022 or height of the Mount Everest in meters. >

> Now, can you define what a "determinable number" is? All I've heard so far are excuses to why you can't.

If you like, here's his latest "explanation" he gave some days ago in
de.sci.mathematik (translated by Google-translator):

|[WM]"Math realism: There are only numbers that can be represented in
| some way. Between 0 and 10^(10)^(100) many are missing, not because
| they are dark, but because they do not exist in our universe.
| For smaller systems like the pocket calculator, many are missing
| between 0 and 10^20.
| Ideal mathematics: There are all numbers between 0 and a number that
| can be specified in some way, such as 10^10^10^10^10 etc. ℕ is
| potentially infinite. Larger numbers are created, expanding the range.
| Actually infinite mathematics: The numbers of ideal mathematics form a
| small area in the dark sea of numbers. Numbers are not created but are
| all there; they are only made accessible. The actually infinite
| sequence consists of the potentially infinite sequence of defined
| numbers followed by obscure numbers, many of which are definable but
| ℵ remain undefinable, for example the last ones before the redefined
| limit. There are no gaps in the real numbers, but there are dark ones
| between each pair of defined real numbers ℵ."

He can't deal with infinity, so he tries to "darken" it...

"Discussing" with WM, one should always keep in mind:
WM is not talking about mathematics but his private nonsense.

Dieter Heidorn

[Fritz Feldhase]

On Sunday, July 30, 2023 at 5:59:52 PM UTC+2, Dieter Heidorn wrote:

> "Discussing" with WM, one should always keep in mind:
> WM is not talking about mathematics but his private nonsense.

Note that the notions/terms always change. Now he's referring to "determinable" numbers, in other cases he's talking about "definable" numbers or "defined" vs. "undefined" numbers. In addition there are "dark" numbers (in contrast to "visible" numbers). etc. etc.

> He can't deal with infinity, so he tries to "darken" it...

Indeed. Or it's just the darkness (lack of light) within him, which ...

[Fritz Feldhase]

On Sunday, July 30, 2023 at 12:24:54 AM UTC+2, WM wrote nonsense:

> Fritz Feldhase schrieb am Samstag, 29. Juli 2023 um 20:52:29 UTC+2:
> >
> > img(NUF) = {0, aleph_0}
> >
> Nonsense.

Nope. Trivially true. Hint: Ax e IR, x <= 0: NUF(x) = 0 and Ax e IR, x > 0: NUF(x) = aleph_0. qed (since dom(NUF) is IR).

> ℵo unit fractions and their internal distances occupy an interval D larger than 0.

This claim doesn't make any sense. Especially in the context of the following claim:

> ∀x ∈ (D, 1]: NUF(x) = ℵo is true.

[Dieter Heidorn]

Fritz Feldhase schrieb:

As we have seen in dsm, he can't no longer deny the fact concerning the
set UF of unit fractions:

∀ n∈ℕ: card( UF\{1/1, 1/2, 1/3, ..., n} ) = ℵo .

His new found emergency exit seems to be that he assumes the following
distribution of unit fractions:

0 D x 1
---|********|-------------------------|--------------------|
|----|---| ℵo UFs 1/n finite
| number of UFs
|
"the very last
ℵo UFs"

Moving from D in direction zero one passes (in WMs universe) the
"very last ℵo UFs" such that the number of UFs between the actual
x-position 0 < x < D and zero decreases in steps of 1 UF respectively.

Just my 2 cents...

Dieter Heidorn

[markus...@gmail.com]

Define "communicate" and "individual".

There is nothing wrong with set theory. The definition is very clear: a set is whatever satisfies the conditions for set. These conditions are summerized as the ZFC axioms.

[markus...@gmail.com]

Seems like he lacks any form of formal training in mathematics. In mathematics, precise and formal definitions are important because without them people don't know what you're talking about.

[Jim Burns]

On 7/30/2023 11:22 AM, WM wrote:
> markus...@gmail.com schrieb am Sonntag,
> 30. Juli 2023 um 16:11:57 UTC+2:

>> I'm not unable to give
>> a formal definition of what a set is.
>> A set is any object that satisfies
>> the axioms of whenever set theory
>> you are dealing with.
>> A ZFC-set is an object that satisfies
>> the ZFC axioms.
>
> What is an object that satisfies the axioms?

We don't need to say what it is.

It is enough that
it is an object that satisfies the axioms.
The axioms are true claims about
what the axioms refer to.

True claims can be augmented, using only
visibly not-first-false claims.

Augmented in that way, only-not-first-falsely,
each descriptive and augmenting claim
is not-first-false.

Description plus augmentation are
a finite sequence of claims.
In a finite sequence of claims,
if each claim is not-first-false,
then each claim is not-false.

The descriptive claims are not-false,
of course.
The only-not-first-false augmenting claims
are also not-false,
and we know they are not-false
without knowing what they refer to,
but by their being only-not-first-false
in that sequence.

> What is an object that satisfies the axioms?

We don't need to say what it is.

That explains the counter-intuitive power
we finite beings have to know about
infinitely-many, almost all of which
we cannot even in principle interact with.

We know the augmenting claims are true
by examining them for not-first-falsity,
not by examining what they refer to.
It doesn't matter what they refer to.
It doesn't matter if we can't,
even in principle, examine what they refer to.
It us by _that claim-sequence_ that we know.

> The concept of infinity from Greeks to Gauss
> differs from actual infinity.

Your actual-infinityᵂᴹ requires the existence of
things which do not equal themselves.

You reject actual-infinityᵂᴹ or you should.

But that doesn't reject _our work_
Actual-infinityᵂᴹ is not _our work_

[WM]

markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 19:24:26 UTC+2:
> söndag 30 juli 2023 kl. 17:23:05 UTC+2 skrev WM:
> > markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 16:11:57 UTC+2:
> > > söndag 30 juli 2023 kl. 16:02:18 UTC+2 skrev WM:
> >
> > > I'm not unable to give a formal definition of what a set is. A set is any object that satisfies the axioms of whenever set theory you are dealing with. A ZFC-set is an object that satisfies the ZFC axioms.
> > What is an object that satisfies the axioms?
> > >
> > > A very precise and concrete definition.
> > No.
> > >
> > > Now, can you define what a "determinable number" is? All I've heard so far are excuses to why you can't. Not a good start. 🤨
> > A determinable number is a number that can be communicated as an individual.
> > >
> > > Classical mathematics, as defined as the mathematics you saw in ancient Greece, did indeed have the concept of infinity.
> > The concept of infinity from Greeks to Gauss differs from actual infinity.
> >

> Define "communicate" and "individual".

If you don't know these words you should repeat school.

>
> There is nothing wrong with set theory.

That is because its adherents refuse to understand what could show set theory wrong. The simplest example: Infinitely many unit fractions and their internal distances occupy a length D > 0. Nevertheless ZF claims that between every x > 0 and 0 there are infinitely many unit fractions:

∀x ∈ (0, 1]: NUF(x) = ℵo

That is impossible because it can be true only for

∀x ∈ (D, 1]: NUF(x) = ℵo

> The definition is very clear: a set is whatever satisfies the conditions for set.

Communicated can be whatever satisfies the conditions for being communitcated.
These conditions are well known from the internet.

Regards, WM

[markus...@gmail.com]

söndag 30 juli 2023 kl. 17:23:05 UTC+2 skrev WM:

A set can be anything you like, as long as it have the properties that the ZFC axioms require. There are many models of set theory.

[markus...@gmail.com]

What are the conditions for "being communicated"? I can list all the ZFC requirements with logical notation. Can you give a definition as precise and exact as the ZFC axioms are stated?

[WM]

Fritz Feldhase schrieb am Sonntag, 30. Juli 2023 um 18:23:45 UTC+2:
> On Sunday, July 30, 2023 at 12:24:54 AM UTC+2, WM wrote nonsense:

> > ℵo unit fractions and their internal distances occupy an interval D larger than 0.
> This claim doesn't make any sense.

It makes sense. The interval (0, D) is abbrevited by D. If you are unable to understand it, try to understand that even two unit fractions and their distance occupy more than one point.

Especially in the context of the following claim:
> > ∀x ∈ (D, 1]: NUF(x) = ℵo is true.

> D *obviously* is a real number here (since it refers as an endpoint of an interval).

It is the endpoint of the interval (0, D).

>
> (Actually, for each and every real number D > 0 ℵo unit fractions are smaller than D (but larger than 0).)

∀x ∈ (0, 1]: NUF(x) = ℵo is wrong at least for the points occupied by the first unit fractions.

Regards, WM

[WM]

Dieter Heidorn schrieb am Sonntag, 30. Juli 2023 um 18:24:20 UTC+2:

> Moving from D in direction zero one passes (in WMs universe) the
> "very last ℵo UFs" such that the number of UFs between the actual
> x-position 0 < x < D and zero decreases in steps of 1 UF respectively.

Obviously NUF(x) cannot increase from 0 to more than 1 in one point because in every point at most one unit fraction exists.

Regards, WM

[WM]

markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 19:52:58 UTC+2:
> söndag 30 juli 2023 kl. 18:15:57 UTC+2 skrev Fritz Feldhase:
> In mathematics, precise and formal definitions are important because without them people don't know what you're talking about.

Intelligent people know it. How long has mathematics existed without the formal crutches?

Regards, WM

[mitchr...@gmail.com]

On Saturday, July 29, 2023 at 3:11:24 PM UTC-7, WM wrote:
> mitchr...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 00:00:43 UTC+2:
> > On Saturday, July 29, 2023 at 2:57:35 PM UTC-7, WM wrote:
> > > mitchr...@gmail.com schrieb am Samstag, 29. Juli 2023 um 18:58:21 UTC+2:

> > > > On Saturday, July 29, 2023 at 4:00:28 AM UTC-7, WM wrote:
> > > > > It is shown by the discreteness of unit fractions that not every real number can be determined.

> > > > Do they not exist in concept?
> > > They exist if Cantor's actual infinity is true. Otherwise they do not exist.
> > And where is the proof of your math?
> It is this: ℵo unit fractions and their internal distances occupy an interval D larger than 0.

> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0.

> NUF(0) = 0.
> Therefore

> ∀x ∈ (0, 1]: NUF(x) = ℵo

> is wrong whereas

> ∀x ∈ (D, 1]: NUF(x) = ℵo is true.
>

> Regards

Can you explain? Something of proof is missing.

[WM]

Jim Burns schrieb am Sonntag, 30. Juli 2023 um 19:57:19 UTC+2:
> On 7/30/2023 11:22 AM, WM wrote:

> > What is an object that satisfies the axioms?
> We don't need to say what it is.
>
> It is enough that
> it is an object that satisfies the axioms.

Typical. Is jhghzgl such an object?

> The axioms are true claims about
> what the axioms refer to.

But they lead to self-contradictions

Inclusion-monotonic sequence of infinite sets cannot have an empty intersection.
∀x ∈ (0, 1]: NUF(x) = ℵo is wrong at least for the first unit fractions.
Bob cannot disappear.

> Actual-infinityᵂᴹ is not _our work_

It is although you don't understand it.

Regards, WM

[WM]

markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 20:02:53 UTC+2:
> söndag 30 juli 2023 kl. 17:23:05 UTC+2 skrev WM:

> > A determinable number is a number that can be communicated as an individual.
> > >
> > > Classical mathematics, as defined as the mathematics you saw in ancient Greece, did indeed have the concept of infinity.
> > The concept of infinity from Greeks to Gauss differs from actual infinity.

> A set can be anything you like, as long as it have the properties that the ZFC axioms require. There are many models of set theory.

A detereminable number can be any number you can deteremine, choose, like.

Regards, WM

[WM]

> What are the conditions for "being communicated"? I can list all the ZFC requirements with logical notation. Can you give a definition as precise and exact as the ZFC axioms are stated?

You might be proud on your use of crutches. I don't need them in oder to think straight. If you need crutches, then apply them and translate the text in your language.

A number can be communicated if you send it and ask for return and receive the same number back.

Regards, WM

[Chris M. Thomasson]

On 7/29/2023 11:53 AM, Fritz Feldhase wrote:
> On Saturday, July 29, 2023 at 8:50:15 PM UTC+2, Chris M. Thomasson wrote:
>> On 7/29/2023 11:46 AM, Fritz Feldhase wrote:
>>> On Saturday, July 29, 2023 at 7:51:33 PM UTC+2, Chris M. Thomasson wrote:
>>>
>>>> Can sqrt(2) be constructed?
>>>
>>> Yeah, by ruler and a compass.
>> Right. Draw the unit square, the diagonal is sqrt(2). :^)
>>
>> I was wondering if The King bassam karzeddin thought that sqrt(2) was
>> some impossible number.
>
> I c. Who knows...

Scary.

[Chris M. Thomasson]

On 7/30/2023 4:19 AM, WM wrote:
> markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 09:33:11 UTC+2:
>> söndag 30 juli 2023 kl. 00:21:38 UTC+2 skrev WM:
>>> markus...@gmail.com schrieb am Samstag, 29. Juli 2023 um 20:38:50 UTC+2:
>>>> lördag 29 juli 2023 kl. 15:27:41 UTC+2 skrev WM:
>>>>> markus...@gmail.com schrieb am Samstag, 29. Juli 2023 um 14:34:40 UTC+2:
>>>>>> lördag 29 juli 2023 kl. 13:00:28 UTC+2 skrev WM:

>>>>>>> It is shown by the discreteness of unit fractions that not every real number can be determined.
>>>>>>>

>>>>>>> Define the function NUF(x) measuring the Number of Unit Fractions between 0 and x. Clearly NUF(x) = 0 for x ≤ 0 and NUF(x) = ℵ0 for every x > 0 that can be determined. But not for all x!
>>>>>>>
>>>>>>> The function NUF(x) is a step-function. It can increase from 0 at x = 0 to greater values, either in a step of size 1 or in a step of size more than 1. But increase by more than 1 is excluded by the gaps between unit fractions:
>>>>>>>

>>>>>>> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0.
>>>>>>>

>>>>>>> Note the universal quantifier, according to which never (and in no limit) two unit fractions occupy the same point x. Therefore the step size can only be 1, resulting in a real x with NUF(x) = 1. This point x however, and all points where NUF(x) < ℵ0, cannot be determined.
>>>>>>>
>>>>>> What does NUF(x) = ℵ0 mean?
>>>>> NUF(x) = ℵ0 means that the number of unit fractions between 0 and x is actually infinite, ℵ0.

>>>>> What's a determinable number?
>>>>> A determinable number is a number that can be communicated between a sender and a receiver such that both know the quantity described by that number. Examples: 7 or π or greatest prime number known in 2022 or height of the Mount Everest in meters.
>>>>>

>>>> What is the domain and codomain of NUF?
>>> Domain is ℝ, co-domain is [0, ℵo]
>>>>
>>>> I don't think your definition of a "determinable number" makes much mathematical sense.
>>> Determinable numbers are the basis of classical mathematics. Determinability is so natural there that it is never mentioned.
>>>
>> Okay, but can you provide a MATHEMATICAL FORMAL DEFINITION for what s determinable number is?
>
> In classical mathematics only determinable numbers are numbers.

[...]

I am thinking of a number that you do not know. In your mind it does not
exist, right? lol.

[markus...@gmail.com]

You just choose different words. What does it mean FORMALLY? If you are going to carry out a proof, you must make sure you're definitions are precise so we all know what you mean.

[markus...@gmail.com]

The problem is that natural language is ambiguous. Formal language isn't. Moreover, you still haven't been able to provide a MATHEMATICAL FORMAL DEFINITION of the word.

I conclude you are unwilling to do and don't care about mathematics.

[FromTheRafters]

WM wrote on 7/30/2023 :
> markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 16:11:57 UTC+2:
>> söndag 30 juli 2023 kl. 16:02:18 UTC+2 skrev WM:
>
>> I'm not unable to give a formal definition of what a set is. A set is any
>> object that satisfies the axioms of whenever set theory you are dealing

>> with. A ZFC-set is an object that satisfies the ZFC axioms.

>
> What is an object that satisfies the axioms?

A "set".

[Jim Burns]

On 7/30/2023 2:36 PM, WM wrote:
> Jim Burns schrieb am Sonntag,
> 30. Juli 2023 um 19:57:19 UTC+2:
>> On 7/30/2023 11:22 AM, WM wrote:

>>> What is an object that satisfies the axioms?
>>
>> We don't need to say what it is.
>> It is enough that
>> it is an object that satisfies the axioms.
> Typical.
> Is jhghzgl such an object?

We don't need to say.

If jhghzgl satisfies the axioms,
then there is vast treasury of
further claims we make about jhghzgl.

We know each of those claims,
because they are not-first-false
augmentation of the axioms.

If jhghzgl doesn't satisfy the axioms,
we aren't making any claims for jhghzgl.

You can't refute no claims.

There is a name for pretending that
we are making claims which we aren't:
the straw-man fallacy.
https://en.wikipedia.org/wiki/Straw_man

>> The axioms are true claims about
>> what the axioms refer to.
>
> But they lead to self-contradictions

If that were true,
the axioms would not refer to anything.

The reason that we prove
for various axiom systems that
things exist to which the axioms might refer
is in order for us to know that
the axioms _do not_ lead to
self-contradiction.

However,
what you list are not self-contradictions.

They are counter-your-intuition,
no more than that.

> Inclusion-monotonic sequence of
> infinite sets cannot have
> an empty intersection.
>
> ∀x ∈ (0, 1]: NUF(x) = ℵo is wrong
> at least for the first unit fractions.
>
> Bob cannot disappear.
>
>> Actual-infinityᵂᴹ is not _our work_
>
> It is although you don't understand it.

In _our_ work
things always equal themselves,
unlike with actual-infinityᵂᴹ

[markus...@gmail.com]

jhghzgl may or may not be a such object. Does it satisfy the axioms?

[WM]

Every number that can be thought of as an individual is determinable.
Every number that is thought of as an individual is determined.

Regards, WM

[WM]

markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 21:50:27 UTC+2:
> söndag 30 juli 2023 kl. 20:38:02 UTC+2 skrev WM:

> > A detereminable number can be any number you can deteremine, choose, like.
> >

> You just choose different words. What does it mean FORMALLY? If you are going to carry out a proof, you must make sure you're definitions are precise so we all know what you mean.

You all know what it means to choose a number or to find a number as the root of an equation or to define a number, for instance by Dedekind-cuts. And those who don't know that are not addressed by my writings.

Regards, WM

[WM]

markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 21:52:34 UTC+2:

> The problem is that natural language is ambiguous. Formal language isn't.

Formal language is based on the means necessary to learn it. Therefore it is as ambiguous as other languages.

"The curse of the invasion of mathematics by mathematical logic is that now any proposition can be represented in a mathematical symbolism, and this makes us feel obliged to understand it. Although of course this method of writing is nothing but the translation of vague ordinary prose." [L. Wittgenstein: "Remarks on the foundations of mathematics", Wiley-Blackwell (1991)]

> Moreover, you still haven't been able to provide a MATHEMATICAL FORMAL DEFINITION of the word.
>
> I conclude you are unwilling to do and don't care about mathematics.

I do mathematics. Formalism is only one of many possible languages. "'Mathematical logic' has completely deformed the thinking of mathematicians and of philosophers, by setting up a superficial interpretation of the forms of our everyday language as an analysis of the structures of facts. Of course in this it has only continued to build on the Aristotelian logic." [L. Wittgenstein: "Remarks on the foundations of mathematics", Wiley-Blackwell (1991)]

Regards, WM

[WM]

Obviously only circular definitions are possible.

Regards, WM

[WM]

markus...@gmail.com schrieb am Montag, 31. Juli 2023 um 10:35:11 UTC+2:
> söndag 30 juli 2023 kl. 20:36:32 UTC+2 skrev WM:
> > Jim Burns schrieb am Sonntag, 30. Juli 2023 um 19:57:19 UTC+2:
> > > On 7/30/2023 11:22 AM, WM wrote:
> >
> > > > What is an object that satisfies the axioms?
> > > We don't need to say what it is.
> > >
> > > It is enough that
> > > it is an object that satisfies the axioms.
> > Typical. Is jhghzgl such an object?

> jhghzgl may or may not be a such object. Does it satisfy the axioms?

That should be determined by the propertied necessary for being a set.

Regards, WM

[WM]

Jim Burns schrieb am Montag, 31. Juli 2023 um 02:39:22 UTC+2:
> On 7/30/2023 2:36 PM, WM wrote:

> However,
> what you list are not self-contradictions.
>
> They are counter-your-intuition,
> no more than that.

They are self-ontradictions. The endsegmente keep infinitely many numbers but their intersections gets empty. Where remain the infinitely many numbers?

> > Inclusion-monotonic sequence of
> > infinite sets cannot have
> > an empty intersection.
> >
> > ∀x ∈ (0, 1]: NUF(x) = ℵo is wrong
> > at least for the first unit fractions.

An increase happens from 0 to ℵo. It cannot happen at one point, according to mathematics. ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 is mathematics, not intuition.
> >
> > Bob cannot disappear.

By simplest logic: In lossless exchange no loss can happen.

> >
> >> Actual-infinityᵂᴹ is not _our work_
> >
> > It is although you don't understand it.
> In _our_ work
> things always equal themselves,
> unlike with actual-infinityᵂᴹ

Things always equal themselves, but we cannot know all of them. We cannot know the first unit fractions. But they must exist if infinite sets exist at all. The latter however is questionable.

Regards, WM

[bassam karzeddin]

As Sir. John Gabriel claim, AP, ..., always, that mathematics is basically a mere discovery & not any human invention

But the global old & modern confusions among ALL acadmic people (especially the mathematicians) is that they can't distinguish between the current eingineering mathematics that are based solely on eingineering human axioms & completely protected by scientists & eingineers who aren't generally any true mathematians
Where they aren't plamed at all in their orientation inorder to solve their little earthy practical problems by many means of endless approximations

For more of illustration for the academic mainstreams mathematicians & alike, I would repeat this practical example about a skilled carpenter (say even before BC), where when asked to make a cube of two units volume, then certainly, he could very easily do the task exactly like any expert mathematicians

Where, nither the skilled carpenters nor the academic mainstreams mathematicians or all other educated humans can simply understand the problem in depth, where this task is actually impossible by any tools or means of endless approximations, since this is actually the same old impossible construction problems raised by the ancient Greeks thousands of years back & still missed theoretically by ALL humans
up to date even with thousands of rigorous elementary proofs

So, the Greek's three impossible construction problems by any tools are infact & purely becoming physiological incurable problems due to the huge mental retardation that had been added to human minds either innocently or so devilishly most likely
But so unfortunately, physiologists can't understand anything also about these old rooted problems nor the mathematicians themselves

The hope only lies upon the future independent & so advanced Artificial intelligence that doesn't have any human traits or features 🤔
Bassam Karzeddin

t
E

[markus...@gmail.com]

All real numbers are Dedekind cuts. So then all real numbers are "determinable"?

[markus...@gmail.com]

What does that mean FORMALLY?

[WM]

bassam karzeddin schrieb am Montag, 31. Juli 2023 um 12:13:01 UTC+2:
> On Monday, July 31, 2023 at 12:42:49 PM UTC+3, WM wrote:

> As Sir. John Gabriel claim, AP, ..., always, that mathematics is basically a mere discovery & not any human invention

Maybe. Actually infinite sets cannot be a human invention.

>
> So, the Greek's three impossible construction problems by any tools

That is not true. They are inaccessible by Plato's arbitrary rule of ruler and compasses. By curves like the quadratrix or Eratosthenes squares they can be solved.

Regards, WM

[WM]

> What does that mean FORMALLY?

Translate it into your language if you need that language to be able to think about things.

A simple example: Natural numbers are accessible, definable, determinable, etc. if they adhere to Peano's axioms, i.e., if they have finite initial segments:
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
but those numbers which can onl be handled collectively are dark:
|ℕ \ {1, 2, 3, ...}| = 0

Certainly you can yourself define formally the natural numbers by the Peano axioms.

Regards, WM

[FromTheRafters]

WM formulated on Monday :

Nothing circular about that. A set is defined by the axioms of the set
theory being discussed.

[WM]

markus...@gmail.com schrieb am Montag, 31. Juli 2023 um 18:50:04 UTC+2:

> All real numbers are Dedekind cuts.

Maybe. But most cuts you cannot find.

> So then all real numbers are "determinable"?

By simple lack of ressources you cannot determine more than countably many numbers.
But even that task cannot be completed.

Regards, WM

[WM]

That is circular and not exclusive. Also not existing sets obey the axioms.
Define a set without reference to the axioms.

Regards, WM

[Jim Burns]

On 7/31/2023 6:06 AM, WM wrote:
> Jim Burns schrieb am Montag,
> 31. Juli 2023 um 02:39:22 UTC+2:

>> However,
>> what you list are not self-contradictions.
>>
>> They are counter-your-intuition,
>> no more than that.
>
> They are self-ontradictions.
> The endsegmente keep infinitely many numbers
> but their intersections gets empty.
> Where remain the infinitely many numbers?

Yes,
their intersection is empty.

Each split F∥E ⇉ ⋃{FISON} is into
a FISON F = ⟨0…i⟩ and
an end segment E = ⟨i⁺¹…⟩

Each number n ∈ ⋃{FISON}
is in a FISON ⟨0…i⟩ and
is not-in an end segment ⟨i⁺¹…⟩

Each number n not-in an end segment ⟨i⁺¹…⟩
is not-in their intersection ⋂{end}

Each number n ∈ U{FISON}
is not-in their intersection ⋂{end}
thus
their intersection ⋂{end} is empty.

> The endsegmente keep infinitely many numbers

Yes,
each end segment is infinite.

Each split F∥E ⇉ ⋃{FISON} is into
a FISON F = ⟨0…i⟩ and
an end segment E = ⟨i⁺¹…⟩

⟨0…i⟩ is 1×1 2-ended: finite.
⟨i⁺¹…⟩ is 1×1 1-ended: infinite.

No mₓ is last
in ⋃{FISON} or in any ⟨i⁺¹…⟩
because
⟨0…mₓ⟩ is followed by ⟨0…mₓ,mₓ⁺¹⟩
and so mx is followed by mx⁺¹
in ⋃{FISON} or in any ⟨i⁺¹…⟩

>>> Inclusion-monotonic sequence of
>>> infinite sets cannot have
>>> an empty intersection.
>>>
>>> ∀x ∈ (0, 1]: NUF(x) = ℵo is wrong
>>> at least for the first unit fractions.
>
> An increase happens from 0 to ℵo.
> It cannot happen at one point,

It happens at zero

in each neighborhood N₀ of 0
there is a ℵ₀-many 1×1 1-ended ⅟n-sequence.

It is not the same sequence in
each neighborhood.

there is no ℵ₀-many 1×1 1-ended ⅟n-sequence
in each neighborhood N₀ of 0

Only 0 is in each neighborhood N₀ of 0

> according to mathematics.
> ∀n ∈ ℕ:
> 1/n - 1/(n+1) = 1/(n(n+1)) > 0
> is mathematics, not intuition.

0 is not a unit fraction.
¬∃n ∈ ℕ⁺: ⅟n = 0

Your formula about unit fractions
does not apply.
It's too bad you (WM) never took
any physics classes.
You would have learned that
formulas can't be used randomly,
anywhere the mood takes you.

>>> Bob cannot disappear.
>
> By simplest logic:
> In lossless exchange no loss can happen.

For each cell p/q
there is a last lossless swap p/q↔n/1
p/q↔n/1 leaves an X in p/q

If Bob Not-An-X is in p/q
then lossless p/q↔n/1 is not swapped.

B⇒¬S
If Bob Not-An-X is in any cell
then not all lossless swaps are swapped.

S⇒¬B
If all lossless swaps are swapped
then Bob Not-An-X is not in any cell.

>>>> Actual-infinityᵂᴹ is not _our work_
>>>
>>> It is although you don't understand it.
>>
>> In _our_ work
>> things always equal themselves,
>> unlike with actual-infinityᵂᴹ
>
> Things always equal themselves,

Then,
each set A can match itself
A ⟷ⁱᵈ A
id(x) = x

Suppose set B contains
a proper subset C which contains
a proper subset D
B ⊃≠ C ⊃≠ D
and C can match D
C ⟷ᶠ D

Then
B contains a proper subset which
B can match.
and
no actually-infiniteᵂᴹ
not-matching-proper-subset but
proper-subset-matching-its-proper-subset
set exists.

Respectively,
C and B\C can match D and B\C
C ⟷ᶠ D
B\C ⟷ⁱᵈ B\C

and
B can match D∪(B\C), a proper subset.
B ⟷ᵍ D∪(B\C)
g(x) = f(x) if x ∈ C
g(x) = x if x ∈ B\C

> but we cannot know all of them.

Knowing all of them isn't needed
in order to know something about
each one of them.

I know that each natural number has
a unique prime factorization.
And yet, I do not know infinitely-many
natural numbers.
There is no conflict between those claims.

> We cannot know the first unit fractions.
> But they must exist
> if infinite sets exist at all.

No!
The opposite of that!

If infinite ordered sets exist,
there must exist ordered sets
without a first or without a last.

| 3. (Paul Stäckel) S can be given a total
| ordering which is well-ordered both forwards
| and backwards. That is, every non-empty
| subset of S has both a least and a greatest
| element in the subset.
|
https://en.wikipedia.org/wiki/Finite_set

You call a finite set "infinite" and
-- surprise! -- you get "contradictions".

> The latter however is questionable.

The sequence of
1×1 2-ended ⟨0…n⟩ ⁺¹ing from 0 is
1×1 1-ended
because
each ⟨0…n⟩ is followed by ⟨0…n,n⁺¹⟩
no ⟨0…n⟩ is a second end.

Question that. <shrug>

[WM]

Jim Burns schrieb am Montag, 31. Juli 2023 um 19:54:19 UTC+2:
> On 7/31/2023 6:06 AM, WM wrote:
> > Jim Burns schrieb am Montag,

> > Where remain the infinitely many numbers?
> Yes,
> their intersection is empty.

Every endsegment that contains at least one number together with E(1) cannot cause an empty intersection. For an empty intersection all numbers must disappear. They cannot disappear in different endsegments. Therefore thy must disappear in one endsegment.

Theorem: The intersection of an infinite sequence of inclusion monotonic infinite sets is infinite.

> > according to mathematics.
> > ∀n ∈ ℕ:
> > 1/n - 1/(n+1) = 1/(n(n+1)) > 0
> > is mathematics, not intuition.
> 0 is not a unit fraction.

Therefore the unit fractions sit at point x > 0. Never more than one at a point.

> ¬∃n ∈ ℕ⁺: ⅟n = 0

That is true only for definable unit fractions.

>
> Your formula about unit fractions
> does not apply.

∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 applies in mathematics without exception.

> >>> Bob cannot disappear.
> >
> > By simplest logic:
> > In lossless exchange no loss can happen.
> For each cell p/q

Irrelevant. In lossless exchange no loss can happen.

> > Things always equal themselves,
> Then,
> each set A can match itself
> A ⟷ⁱᵈ A
> id(x) = x

Of course. But most x cannot be identified let alone mapped to other elements.

Regards, WM

[Fritz Feldhase]

On Monday, July 31, 2023 at 8:29:21 PM UTC+2, WM wrote:

> For an empty intersection

for each and every natural number n there must be an endsegment E such that n is not an element in E.

> They cannot ["]disappear["] in different endsegments.

A rather "unlucky formulation" (i.e. Mückenheim nonsense).

Actually, each and every natural number n is in E(n) but NOT in E(n+1) (and not in E(n+2), E(n+3), ...).

In this sense the natural number (first) "disappear" in different endsegements:

1 in E(2)
2 in E(3)
:
n in E(n+1)
:

[Jim Burns]

On 7/31/2023 2:29 PM, WM wrote:
> Jim Burns schrieb am Montag,
> 31. Juli 2023 um 19:54:19 UTC+2:
>> On 7/31/2023 6:06 AM, WM wrote:

>>> according to mathematics.
>>> ∀n ∈ ℕ:
>>> 1/n - 1/(n+1) = 1/(n(n+1)) > 0
>>> is mathematics, not intuition.
>>
>> 0 is not a unit fraction.
>
> Therefore
> the unit fractions sit at point x > 0.
> Never more than one at a point.
>
>> ¬∃n ∈ ℕ⁺: ⅟n = 0
>
> That is true only for definable
> unit fractions.

¬∃n ∈ ℕ⁺: 1 = n⋅0

because
n⋅0 = 0

n⋅m⁺¹ = n⋅m + n

⅟n = 0 ⟺ 1 = n⋅0

[mitchr...@gmail.com]

If a quantity can not be determined how do you use it?

[Chris M. Thomasson]

But you cannot think of my number. So, its dark to you, it will never
exist to you, even though its there. You dark numbers are kind of funny
to me in certain ways.

[FromTheRafters]

Fritz Feldhase laid this down on his screen :

> On Monday, July 31, 2023 at 8:29:21 PM UTC+2, WM wrote:
>
>> For an empty intersection
>
> for each and every natural number n there must be an endsegment E such that n
> is not an element in E.

Not the way WM does it though. His endsegments start with the same
element which ends the FISON.

{1, 2, 3, ... n} {n, n+1, ... }

In his world, it makes sense to him. :)

[FromTheRafters]

WM pretended :

> FromTheRafters schrieb am Montag, 31. Juli 2023 um 19:43:35 UTC+2:
>> WM formulated on Monday :
>>> FromTheRafters schrieb am Montag, 31. Juli 2023 um 00:43:57 UTC+2:
>>>> WM wrote on 7/30/2023 :
>>>>> markus...@gmail.com schrieb am Sonntag, 30. Juli 2023 um 16:11:57 UTC+2:
>>>>>> söndag 30 juli 2023 kl. 16:02:18 UTC+2 skrev WM:
>>>>>> I'm not unable to give a formal definition of what a set is. A set is
>>>>>> any object that satisfies the axioms of whenever set theory you are
>>>>>> dealing with. A ZFC-set is an object that satisfies the ZFC axioms.
>>>>>
>>>>> What is an object that satisfies the axioms?
>>>> A "set".
>>>
>>> Obviously only circular definitions are possible.
>> Nothing circular about that. A set is defined by the axioms of the set
>> theory being discussed.
>
> That is circular and not exclusive.

It is not circular, nor non-exclusive.

> Also not existing sets obey the axioms.

No they don't.

From your playbook:

Name a not existing set, fail.

Hint, non existing things don't exist.

> Define a set without reference to the axioms.

A set is a collection of well-defined distinct objects.

{horse, elephant, jackass}

Of course the notions of "well-defined" and "distinct" come into play.

[Fritz Feldhase]

Yes, I know. Still my claim is true. ( Look again. :-P )

Hint: For any n such an E may be E(n+1) for example.

> In his world, it makes sense to him. :)

Yes, I know. As well as leaving out 0 in IN _in a set theoretic context_. ***sigh***

[Fritz Feldhase]

On Sunday, July 30, 2023 at 9:40:42 PM UTC+2, Chris M. Thomasson wrote:

> I am thinking of a number [...]

Me too!

https://www.youtube.com/watch?v=rVtHrgdcvZA

:-)

[FromTheRafters]

Fritz Feldhase was thinking very hard :

> On Tuesday, August 1, 2023 at 12:17:31 AM UTC+2, FromTheRafters wrote:
>> Fritz Feldhase laid this down on his screen :
>>> On Monday, July 31, 2023 at 8:29:21 PM UTC+2, WM wrote:
>>>>
>>>> For an empty intersection
>>>>
>>> for each and every natural number n there must be an endsegment E such that
>>> n is not an element in E.
>> >
>> Not the way WM does it though. His endsegments start with the same
>> element which ends the FISON.
>>
>> {1, 2, 3, ... n} {n, n+1, ... }
>
> Yes, I know. Still my claim is true. ( Look again. :-P )

Okay, you got me there, there must be infinitely many such endsegments
for each n then.

[Chris M. Thomasson]

YUP!!!! Not to give too much away, it might even contain a number just
for men. lol!!!!!!

https://youtu.be/rVtHrgdcvZA?t=48

AAHAHAHAAH!

[Fritz Feldhase]

On Tuesday, August 1, 2023 at 12:27:23 AM UTC+2, FromTheRafters wrote:

> Hint, non existing things don't exist.

Sounds reasonable.:-)

> A set is a collection of well-defined distinct objects.

or the empty set.

No joke, in (one of) Suppes' book(s) you will find the definition:

| x is a set <-> Ey e x v x = 0 ,

where "0" is a primitive (accompanied by the axiom "~Ex(x e 0)").

> {horse, elephant, jackass}

Right. In this case we have, say, horse e {horse, elephant, jackass}. Hence (from the definition obove) {horse, elephant, jackass} is a set.

(Not just want to b a smart-arse.) There's a serious background here: If we are referring to Cantor's famous definition

"A set is a collection of definite, distinguishable objects of perception or thought conceived as a whole."

we might (be justified to) deny the possibility of an "empty set".

Of course, an empty set is indispensable in (modern) set theory.

[Fritz Feldhase]

On Tuesday, August 1, 2023 at 12:35:53 AM UTC+2, Chris M. Thomasson wrote:
> On 7/31/2023 3:30 PM, Fritz Feldhase wrote:
> > On Sunday, July 30, 2023 at 9:40:42 PM UTC+2, Chris M. Thomasson wrote:
> > >
> > > I am thinking of a number [...]
> > >
> > Me too!
> >
> > https://www.youtube.com/watch?v=rVtHrgdcvZA
> >
> > :-)
> >
> YUP!!!! Not to give too much away, it might even contain a number just for men. lol!!!!!!

:-)

> https://youtu.be/rVtHrgdcvZA?t=48
>
> AAHAHAHAAH!

Yeah, it ***is*** funny.

Too bad, that WM won't get the joke. :-)

[mitchr...@gmail.com]

It goes to show your math is only in your imagination.
Just as the i formula has no solution.
Where are your numbers now?

[Fritz Feldhase]

On Tuesday, August 1, 2023 at 1:03:14 AM UTC+2, mitchr...@gmail.com wrote:

> It goes to show your math is only in your imagination.

Indeed!

> Where are your numbers now?

Nowhere.

[mitchr...@gmail.com]

God is math before man.

[Chris M. Thomasson]

I am looking at my number right now.

>
> Nowhere.

[Fritz Feldhase]

If you say so.:-P

[Chris M. Thomasson]

ROFL!!!

[Fritz Feldhase]

Well, if there is a god ...

[mitchr...@gmail.com]

If there is math before man it is God.

[markus...@gmail.com]

Define what you mean by "find a cut".

[markus...@gmail.com]

It's not circular. Everything that satisfies the ZFC axioms is a ZFC-set. Everything that doesn't do that is not a ZFC-set.

[Jim Burns]

On 7/31/2023 2:29 PM, WM wrote:
> Jim Burns schrieb am Montag,
> 31. Juli 2023 um 19:54:19 UTC+2:
>> On 7/31/2023 6:06 AM, WM wrote:

>>> Where remain the infinitely many numbers?
>>
>> Yes,
>> their intersection is empty.
>
> Every endsegment that contains
> at least one number together with E(1)
> cannot cause an empty intersection.

Each end segment E that contains
at least one number j
does not equal the intersection of all.

That number j is
not-in a different end segment E\⟨1…j⟩
not-in all end segments
not-in their intersection ⋂{end}
_but_ in E

E ∋ j ∉ ⋂{end}
E ≠ ⋂{end}

> For an empty intersection
> all numbers must disappear.

Define ℕ := ⋃{FISON}

∀j ∈ ℕ: j ∉ ℕ\⟨1…j⟩ ∧ j ∉ ⋂{end}

⋂{end} = ∅

> They cannot disappear in
> different endsegments.

No, they can.

[WM]

Fritz Feldhase schrieb am Montag, 31. Juli 2023 um 22:04:14 UTC+2:
> On Monday, July 31, 2023 at 8:29:21 PM UTC+2, WM wrote:
>
> > For an empty intersection
>
> for each and every natural number n there must be an endsegment E such that n is not an element in E.

Then no number remains.

1) As long as an endsegment contains an element, it has this element together with all predecessors. If all endsegements contain elements, then all have them together with all predecessors. If all endsegements contain elements, then there are no successors deviating from this rule.

2) Infinite endsegments need infinitely many numbers for their contents. Therefore only finitely many numbers can act as their indices. Hence there are not infinitely many infinite endsegments. Every intersection of finitely many endsegments is infinite.

Regards, WM

[WM]

mitchr...@gmail.com schrieb am Montag, 31. Juli 2023 um 22:45:14 UTC+2:

> If a quantity can not be determined how do you use it?

It cannot be used as an individual. But we can use the whole collection.
Example: Every n that can be determined, has ℵo successors
∀n ∈ ℕ_det: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
Collectively all n can be used such that no successors remain
|ℕ \ {1, 2, 3, ...}| = 0 .

Regards, WM

[WM]

Chris M. Thomasson schrieb am Dienstag, 1. August 2023 um 00:13:30 UTC+2:
> On 7/31/2023 2:36 AM, WM wrote:
> > Chris M. Thomasson schrieb am Sonntag, 30. Juli 2023 um 21:40:42 UTC+2:
> >> On 7/30/2023 4:19 AM, WM wrote:
> >
> >>> In classical mathematics only determinable numbers are numbers.
> >> [...]
> >>
> >> I am thinking of a number that you do not know. In your mind it does not
> >> exist, right? lol.
> >
> > Every number that can be thought of as an individual is determinable.
>
> > Every number that is thought of as an individual is determined.
> But you cannot think of my number. So, its dark to you,

It would be dark if I could not define it. It is not dark as long as it is within reach, i.e., smaller than the greatest number I am able to define. Considering the whole system earth your number is crtainly not dark as long as anyone is able to define or communicate it.

Regards, WM

[Chris M. Thomasson]

On 8/1/2023 1:07 PM, WM wrote:
> Chris M. Thomasson schrieb am Dienstag, 1. August 2023 um 00:13:30 UTC+2:
>> On 7/31/2023 2:36 AM, WM wrote:
>>> Chris M. Thomasson schrieb am Sonntag, 30. Juli 2023 um 21:40:42 UTC+2:
>>>> On 7/30/2023 4:19 AM, WM wrote:
>>>
>>>>> In classical mathematics only determinable numbers are numbers.
>>>> [...]
>>>>
>>>> I am thinking of a number that you do not know. In your mind it does not
>>>> exist, right? lol.
>>>
>>> Every number that can be thought of as an individual is determinable.
>>
>>> Every number that is thought of as an individual is determined.
>> But you cannot think of my number. So, its dark to you,
>
> It would be dark if I could not define it.

Huh! You are a strange person. Its MY number, got it? Well, its dark to
you. Wtf Man!

> It is not dark as long as it is within reach,

HUH!!!!! Its MY NUMBER ASSHOLE!!!!

[...]

[WM]

markus...@gmail.com schrieb am Dienstag, 1. August 2023 um 12:43:32 UTC+2:
> måndag 31 juli 2023 kl. 19:46:50 UTC+2 skrev WM:
> > markus...@gmail.com schrieb am Montag, 31. Juli 2023 um 18:50:04 UTC+2:
> >
> > > All real numbers are Dedekind cuts.
> > Maybe. But most cuts you cannot find.
> > > So then all real numbers are "determinable"?
> > By simple lack of ressources you cannot determine more than countably many numbers.
> > But even that task cannot be completed.
> >

> Define what you mean by "find a cut".

Determine the resulting real number. Write it in any suitable language.

Regards, WM

[Fritz Feldhase]

On Tuesday, August 1, 2023 at 10:10:54 PM UTC+2, Chris M. Thomasson wrote:

> HUH!!!!! Its MY NUMBER ASSHOLE!!!!

*lol* Mind your language, my friend. :-)

[WM]

Jim Burns schrieb am Dienstag, 1. August 2023 um 19:38:13 UTC+2:
> On 7/31/2023 2:29 PM, WM wrote:
> > Jim Burns schrieb am Montag,
> > 31. Juli 2023 um 19:54:19 UTC+2:
> >> On 7/31/2023 6:06 AM, WM wrote:
>
> >>> Where remain the infinitely many numbers?
> >>
> >> Yes,
> >> their intersection is empty.
> >
> > Every endsegment that contains
> > at least one number together with E(1)
> > cannot cause an empty intersection.
> Each end segment E that contains
> at least one number j
> does not equal the intersection of all.

Therefore the empty endsegment ist needed.

>
> > They cannot disappear in
> > different endsegments.
> No, they can.

Like Bob! Not in inclusion monotonic endsegements!! Not in logic!!!

Regards, WM