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A number that cannot be determined

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WM

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Jul 29, 2023, 7:00:28 AM7/29/23
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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

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Jul 29, 2023, 8:34:40 AM7/29/23
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What does NUF(x) = ℵ0 mean? What's a determinable number?

WM

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Jul 29, 2023, 9:27:41 AM7/29/23
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> 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

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Jul 29, 2023, 12:58:21 PM7/29/23
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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

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Jul 29, 2023, 1:28:32 PM7/29/23
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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

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Jul 29, 2023, 1:47:47 PM7/29/23
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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

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Jul 29, 2023, 1:51:33 PM7/29/23
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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

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Jul 29, 2023, 2:38:50 PM7/29/23
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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

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Jul 29, 2023, 2:46:30 PM7/29/23
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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

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Jul 29, 2023, 2:50:15 PM7/29/23
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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

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Jul 29, 2023, 2:52:29 PM7/29/23
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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

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Jul 29, 2023, 2:53:28 PM7/29/23
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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

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Jul 29, 2023, 5:57:35 PM7/29/23
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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

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Jul 29, 2023, 6:00:43 PM7/29/23
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And where is the proof of your math?

> Regards, WM

WM

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Jul 29, 2023, 6:11:24 PM7/29/23
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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

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Jul 29, 2023, 6:21:38 PM7/29/23
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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

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Jul 29, 2023, 6:24:54 PM7/29/23
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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

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Jul 29, 2023, 6:26:56 PM7/29/23
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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

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Jul 29, 2023, 11:34:44 PM7/29/23
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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

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Jul 30, 2023, 3:33:11 AM7/30/23
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Okay, but can you provide a MATHEMATICAL FORMAL DEFINITION for what s determinable number is?

WM

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Jul 30, 2023, 7:20:01 AM7/30/23
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> 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

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Jul 30, 2023, 9:52:46 AM7/30/23
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What is "classical mathematics" and why are you unable to give a precise formal definition of what a "determinable number" is?

WM

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Jul 30, 2023, 10:02:18 AM7/30/23
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> 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

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Jul 30, 2023, 10:11:57 AM7/30/23
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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

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Jul 30, 2023, 11:23:05 AM7/30/23
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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

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Jul 30, 2023, 11:34:03 AM7/30/23
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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

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Jul 30, 2023, 11:36:42 AM7/30/23
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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

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Jul 30, 2023, 11:39:28 AM7/30/23
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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

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Jul 30, 2023, 11:51:58 AM7/30/23
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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

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Jul 30, 2023, 11:59:52 AM7/30/23
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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

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Jul 30, 2023, 12:15:57 PM7/30/23
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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

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Jul 30, 2023, 12:23:45 PM7/30/23
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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

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Jul 30, 2023, 12:24:20 PM7/30/23
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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

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Jul 30, 2023, 1:24:26 PM7/30/23
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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

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Jul 30, 2023, 1:52:58 PM7/30/23
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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

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Jul 30, 2023, 1:57:19 PM7/30/23
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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

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Jul 30, 2023, 1:57:27 PM7/30/23
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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

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Jul 30, 2023, 2:02:53 PM7/30/23
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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

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Jul 30, 2023, 2:06:18 PM7/30/23
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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

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Jul 30, 2023, 2:27:37 PM7/30/23
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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

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Jul 30, 2023, 2:30:22 PM7/30/23
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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

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Jul 30, 2023, 2:32:21 PM7/30/23
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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

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Jul 30, 2023, 2:34:02 PM7/30/23
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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

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Jul 30, 2023, 2:36:32 PM7/30/23
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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

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Jul 30, 2023, 2:38:02 PM7/30/23
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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

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Jul 30, 2023, 2:42:29 PM7/30/23
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> 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

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Jul 30, 2023, 3:39:44 PM7/30/23
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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

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Jul 30, 2023, 3:40:42 PM7/30/23
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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

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Jul 30, 2023, 3:50:27 PM7/30/23
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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

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Jul 30, 2023, 3:52:34 PM7/30/23
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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

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Jul 30, 2023, 6:43:57 PM7/30/23