On 8/21/2021 5:32 PM, WM wrote:
> Jim Burns schrieb
> am Samstag, 21. August 2021 um 20:43:52 UTC+2:
>> On 8/21/2021 9:37 AM, WM wrote:

>>> Fact is that the natural number completing the indexing of
>>> all fractions of the interval (n, n+1]
>>
>> There is no such natural number.
>
> Then the completing is accomplished by belief only?

There are infinitely-many indexed rationals in the
interval (n,n+1], each rational with a finite index.

( Use index(p/q) = q*q*p*p/rad(p) with gcd(p,q) = 1
( ( Assume that there are finitely-many indexed rationals.
( in (n,n+1], each with a finite index.
( Then, ordered in the standard in-the-line way, the
( indexed rationals in (n,n+1] are steppable.
( For any split B,A, there is a crossing-pair p/q,r/s of
( indexed rationals.
( Because crossing-pair, there are no indexed rationals
( between p/q and r/s.
( However, (p/q + r/s)/2 is between p/q and r/s,
( and it has a finite index, index((p/q + r/s)/2).
( Contradiction.
( Therefore, there are infinitely-many indexed rationals
( in (n,n+1], each with a finite index.

All natural numbers are finite.
There is no natural-number/finite-index after infinitely-many
indexed rationals, each with a finite index.

>> It would need to be a finite index after infinitely-many
>> indexes. There is no such natural number.
>
> Completion is claimed as basis of set theory.
> How is it achieved?

It is not achieved with a last natural number, or a last
indexed rational in (n,n+1], each with a finite index.

You (WM) seem to have a problem.

We (broadly) have developed _variables_ by which we may refer
to _one of_ some collection without specifying _which one_

This is very useful for partially describing each individual in
the collection, even if there are infinitely-many individuals,
even if we ourselves are finite. The key here is to NOT SPECIFY
which individual in the collection is referred to and to only make
claims that are TRUE NO MATTER WHICH individual is referred to.

If we proceed from this partial description true of each
individual of interest using _only_ truth-preserving inferences,
we will be able to derive further claims that we will
know to be true of the same individuals, even if there are
infinitely-many individuals, even if we ourselves are finite.

This makes a better basis that what you apparently think
is the basis of "set theory". The absence of a last natural
number or a last indexed rational in (n,n+1], each with a
finite index does not create a problem for this basis.

You (WM) have before you two bases for reasoning.
One has a problem. One does not have that problem.
What do you think you should do?

On 8/21/2021 4:28 PM, WM wrote:
> William schrieb am Samstag, 21. August 2021 um 20:59:41 UTC+2:
>> On Saturday, August 21, 2021 at 2:01:46 PM UTC-4, WM wrote:
>>> William schrieb am Samstag, 21. August 2021 um 18:12:31 UTC+2:
>>>> On Saturday, August 21, 2021 at 9:35:22 AM UTC-4, WM wrote:
>>>
>>>>>> It does not. The *set of FISONs* cannot be analyzed step by step.
>>>>> All FISONs can. But there is no set because it is only a potentially infinite collection
>>>> Piffle. "Potentially Infinite" is nonsense.
>>> Mathematics disregards your personal preferences. If you don't like potential infinity
>> The reason I don't like "potential infinity" is that it is nonsense. It does not exist outside of Wolkenmuekenheim.
>
> There you are very wrong.
> "A potential infinity is a quantity which is finite but indefinitely large. For instance, when we enumerate the natural numbers as 0, 1, 2, ..., n, n+1, ..., the enumeration is finite at any point in time,

*there is your mistake*, *you are using time*, *you are stopping at k*. Of course your counting is finite if you stop at k.

so "potential infinity" = finite, that is all. Stop using misleading words, and show your math.

>but it grows indefinitely and without bound. [...] An actual infinity is a completed infinite totality. Examples: , , C[0, 1], L2[0, 1], etc. Other examples: gods, devils, etc." [S.G. Simpson: "Potential versus actual infinity: Insights from reverse mathematics" (2015)]

the reference paper is not peer reviewed, and is lecture notes, Fail.

>
> An interval of natural numbers without any prime number is called a prime gap. The sequence of prime gaps assumes arbitrarily large intervals but it cannot become actually infinite.

wrong thinking, it is infinite, unless you prove/show it is Finite.

<snip rest of misleading garf>

On 8/21/2021 4:37 PM, WM wrote:
> Python schrieb am Samstag, 21. August 2021 um 21:34:11 UTC+2:
>> WM wrote:
>
>>> (1) When indexing the rationals all unit intervals can be completed.
>> What does that even mean?
>
> That can be answered best by the master himself (the German terminus is vollständig):
>
> "Zwei wohlgeordnete Mengen M und N heissen von gleichem Typus oder auch von gleicher Anzahl, wenn sie sich gegenseitig eindeutig und vollständig unter beidseitiger Wahrung der Rangfolge ihrer Elemente auf einander beziehen, abbilden lassen;" [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]
>
> "Zwei bestimmte Mengen M und M1 nennen wir äquivalent (in Zeichen: M ~ M1), wenn es möglich ist, dieselben gesetzmäßig, gegenseitig eindeutig und vollständig, Element für Element, einander zuzuordnen." [Cantor, p. 412]
>
> "doch gibt es immer viele, im allgemeinen sogar unzählig viele Zuordnungsgesetze, durch welche zwei äquivalente Mengen in gegenseitig eindeutige und vollständige Beziehung zueinander gebracht werden können." [Cantor, p. 413]
>
> "eine solche gegenseitig eindeutige und vollständige Korrespondenz hergestellt [...] irgendeine gegenseitig eindeutige und vollständige Zuordnung der beiden Mengen [...] auch eine gegenseitig eindeutige und vollständige Korrespondenz" [Cantor, p. 415]
>
> "Zwei n-fach geordnete Mengen M und N werden 'ähnlich' genannt, wenn es möglich ist, sie gegenseitig eindeutig und vollständig, Element für Element, einander so zuzuordnen," [Cantor, p. 424]
>
>>> (2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
>> What does that even mean?
>
> A rational number belongs to one and only one unit interval.

Wrong!

Fact, there is no interval small enough to contain only one rational number.

[ More WM crank chumming the waters of sci.math]

>
> Regards, WM
>

On 8/21/2021 4:32 PM, WM wrote:
> Jim Burns schrieb am Samstag, 21. August 2021 um 20:43:52 UTC+2:
>> On 8/21/2021 9:37 AM, WM wrote:
>
>>> Fact is that the natural number completing the indexing of
>>> all fractions of the interval (n, n+1]
>> There is no such natural number.
>
> Then the completing is accomplished by belief only?

you do math by belief only.

>
>> It would need to be a finite index after infinitely-many
>> indexes. There is no such natural number.
>
> Completion is claimed as basis of set theory. How is it achieved?

define "completion"

>
> Regards, WM
>

Sergio wrote:

>>>> Fact is that the natural number completing the indexing of all
>>>> fractions of the interval (n, n+1]
>>> There is no such natural number.
>>
>> Then the completing is accomplished by belief only?
>
> you do math by belief only.

That's the problem with beliefs. You can't arrest Fraudci or bill gaytes,
because are not there. All you see those places are *deep_fake*. Very
easy. Proof:

What’s With the Weird Chins?
https://153news.net/watch_video.php?v=2A3G36WS6M88

because the stole your pension, they want you no more.

They Have STOLEN Your Pension Funds...Now Get The Jab.
https://153news.net/watch_video.php?v=7GXUD52NN2UH

On 8/22/2021 6:42 AM, Ward Ehlers wrote:
> Sergio wrote:
>
>>>>> Fact is that the natural number completing the indexing of all
>>>>> fractions of the interval (n, n+1]
>>>> There is no such natural number.
>>>
>>> Then the completing is accomplished by belief only?
>>
>> you do math by belief only.
>
> That's the problem with beliefs. You can't arrest Fraudci or bill gaytes,
> because are not there. All you see those places are *deep_fake*. Very
> easy. Proof:
>
> What’s With the Weird Chins?
> https://153news.net/watch_video.php?v=2A3G36WS6M88
>
> because the stole your pension, they want you no more.
>
> They Have STOLEN Your Pension Funds...Now Get The Jab.
> https://153news.net/watch_video.php?v=7GXUD52NN2UH
>

Wierd Chin alright, Alien chin for sure

Bankers only steal funds from commie proletariat scum, and give it to the Rich Commies

Sergio wrote:

>> What’s With the Weird Chins?
>> https://153news.net/watch_video.php?v=2A3G36WS6M88
>>
>> because the stole your pension, they want you no more.
>>
>> They Have STOLEN Your Pension Funds...Now Get The Jab.
>> https://153news.net/watch_video.php?v=7GXUD52NN2UH
>
> Wierd Chin alright, Alien chin for sure

not weird, that's *deep_fake*. They are going to correct the algorithm
and in a few days you'll not be able to see anything abnormal.

Don't be that fool, capitalist scumbag. They come after you as the number
one.

On 8/22/2021 9:22 AM, Ward Ehlers wrote:
> Sergio wrote:
>
>>> What’s With the Weird Chins?
>>> https://153news.net/watch_video.php?v=2A3G36WS6M88
>>>
>>> because the stole your pension, they want you no more.
>>>
>>> They Have STOLEN Your Pension Funds...Now Get The Jab.
>>> https://153news.net/watch_video.php?v=7GXUD52NN2UH
>>
>> Wierd Chin alright, Alien chin for sure
>
> not weird, that's *deep_fake*. They are going to correct the algorithm
> and in a few days you'll not be able to see anything abnormal.
>
> Don't be that fool, capitalist scumbag. They come after you as the number
> one.
>

you lumpenproletariats are all alike

we will read James Baldwin and John Rechy alongside queer of color critique to reexamine the intersections of class, sexuality and race might be
rethought through the category of the lumpenproletariat. Then we will turn to John Waters and the trash aesthetic of the 70s – queer lumpens teetering
between hippies and punks. And finally, we will examine contemporary transatlantic crises of civil society and electoral democracy, characterized in
both the US and the UK by populist, authoritarian nativisms opposed to yet complicit with a slick international technocracy. Who, in other words, could
be the committed revolutionary subjects of the future while industrial action continues to be abrogated as “outdated,” and a supposedly uniformly
racist, apparently entirely white, fetishized image of the proletariat is mobilized against a racialized immigrant “underclass”?

Sergio wrote:

>>>> because the stole your pension, they want you no more.
>>>>
>>>> They Have STOLEN Your Pension Funds...Now Get The Jab.
>>>> https://153news.net/watch_video.php?v=7GXUD52NN2UH
>>>
>>> Wierd Chin alright, Alien chin for sure
>>
>> not weird, that's *deep_fake*. They are going to correct the algorithm
>> and in a few days you'll not be able to see anything abnormal.
>> Don't be that fool, capitalist scumbag. They come after you as the
>> number one.
>>
> you lumpenproletariats are all alike

not sure that I am that, but thanks for not calling me a capitalist.

We have big money capitalist fraudci, capitalist 20:1 profit bill gaytes,
big pharma capitalist trumpf, capitalist biden (stole $s in millions from
ukraine, poorest in europe), capitalist soros, capitalist rokafella,
capitalist claus schwab (anal), capitalist pfitzer, capitalist moderna
etc etc and etc. They don't need you, they want you no more.

Also, common misconception, they don't need robots to replace humans,
because there will be nobody wanting buying what they produce.

Hence *mass_production* will *STOP*, all your investings will go directly
into your *dirty_capitalist_ass*.

William schrieb am Samstag, 21. August 2021 um 23:35:28 UTC+2:
> On Saturday, August 21, 2021 at 5:28:38 PM UTC-4, WM wrote:
> > > The reason I don't like "potential infinity" is that it is nonsense. It does not exist outside of Wolkenmuekenheim.
> > There you are very wrong.
> > "A potential infinity is a quantity which is finite but indefinitely large. For instance, when we enumerate the natural numbers as 0, 1, 2, ..., n, n+1, ..., the enumeration is finite at any point in time, but it grows indefinitely and without bound. [...] An actual infinity is a completed infinite totality.
> >
> The use of the word "completed" gives this away. It is nonsense. It changes nothing.

Here is the proof of incomplete infinity:

(1) When indexing the rationals all unit intervals can be completed.
(2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
(3) So there are at least two indices required to complete all intervals.
(4) You cannot find out their order.

Although you prefer not to take notice of this fact, it is nevertheless unavoidable.

Regards, WM

Jim Burns schrieb am Sonntag, 22. August 2021 um 00:24:41 UTC+2:
> On 8/21/2021 5:32 PM, WM wrote:

> > Then the completing is accomplished by belief only?
> There are infinitely-many indexed rationals in the
> interval (n,n+1], each rational with a finite index.

Infinitely many is right, but completed infinity does not exist other than being dark.
Proof:

(1) When indexing the rationals all unit intervals can be completed.
(2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
(3) So there are at least two indices required to complete all intervals.
(4) You cannot find out their order.

Although you prefer not to take notice of this fact, it is nevertheless unavoidable.

> ( Use index(p/q) = q*q*p*p/rad(p) with gcd(p,q) = 1
> (
> ( Assume that there are finitely-many indexed rationals.

That is not the alternative.

> There is no natural-number/finite-index after infinitely-many
> indexed rationals, each with a finite index.

Correct. There is potential infinity. Not possible to complete it. This is proved above.

Look: The mapping of |N can show that there are as many rational numbers in the interval (0, 10^-100000) as in (-oo, oo), as many prime numbers as algebraic numbers or definable real and complex numbers. Don't you suspect an error of the counter-intuitive interpretation?

The explanation is very simple and will convince every student not yet infected too much by set theory: There is a potentially infinite set of definable elements in every infinite set. They can be mapped to each other. That's why all sets have same cardinality.

Regards, WM

On Sunday, August 22, 2021 at 2:02:14 PM UTC-4, WM wrote:
> William schrieb am Samstag, 21. August 2021 um 23:35:28 UTC+2:
> > On Saturday, August 21, 2021 at 5:28:38 PM UTC-4, WM wrote:
>
> > > > The reason I don't like "potential infinity" is that it is nonsense.. It does not exist outside of Wolkenmuekenheim.
> > > There you are very wrong.
> > > "A potential infinity is a quantity which is finite but indefinitely large. For instance, when we enumerate the natural numbers as 0, 1, 2, ..., n, n+1, ..., the enumeration is finite at any point in time, but it grows indefinitely and without bound. [...] An actual infinity is a completed infinite totality.
> > >
> > The use of the word "completed" gives this away. It is nonsense. It changes nothing.
> Here is the proof of incomplete

The term "complete" is nonsense. it implies that there are "incomplete" sets, sets that should contain something that they do not. All sets are "complete", all sets contain exactly what they contain. There are subsets and proper subsets but there is no such thing as an "incomplete" set.

If a "proof" uses the word "complete" it is nonsense.

--
William Hughes

William brought next idea :
> On Sunday, August 22, 2021 at 2:02:14 PM UTC-4, WM wrote:
>> William schrieb am Samstag, 21. August 2021 um 23:35:28 UTC+2:
>>> On Saturday, August 21, 2021 at 5:28:38 PM UTC-4, WM wrote:
>>>>> The reason I don't like "potential infinity" is that it is nonsense. It
>>>>> does not exist outside of Wolkenmuekenheim.
>>>> There you are very wrong.
>>>> "A potential infinity is a quantity which is finite but indefinitely
>>>> large. For instance, when we enumerate the natural numbers as 0, 1, 2,
>>>> ..., n, n+1, ..., the enumeration is finite at any point in time, but it
>>>> grows indefinitely and without bound. [...] An actual infinity is a
>>>> completed infinite totality.
>>>>
>>> The use of the word "completed" gives this away. It is nonsense. It changes
>>> nothing.
>> Here is the proof of incomplete
>
> The term "complete" is nonsense. it implies that there are "incomplete"
> sets, sets that should contain something that they do not. All sets are
> "complete", all sets contain exactly what they contain. There are subsets
> and proper subsets but there is no such thing as an "incomplete" set.
>
> If a "proof" uses the word "complete" it is nonsense.

Q is a field, but not complete. R is a field, and is complete. Complete
here meaning there are no holes (gaps) which we would need to invent
new numbers to fill like the irrationals did for Q.

William schrieb am Sonntag, 22. August 2021 um 20:32:02 UTC+2:
> On Sunday, August 22, 2021 at 2:02:14 PM UTC-4, WM wrote:
> > William schrieb am Samstag, 21. August 2021 um 23:35:28 UTC+2:
> > > On Saturday, August 21, 2021 at 5:28:38 PM UTC-4, WM wrote:
> >
> > > > > The reason I don't like "potential infinity" is that it is nonsense. It does not exist outside of Wolkenmuekenheim.
> > > > There you are very wrong.
> > > > "A potential infinity is a quantity which is finite but indefinitely large. For instance, when we enumerate the natural numbers as 0, 1, 2, ...., n, n+1, ..., the enumeration is finite at any point in time, but it grows indefinitely and without bound. [...] An actual infinity is a completed infinite totality.
> > > >
> > > The use of the word "completed" gives this away. It is nonsense. It changes nothing.
> > Here is the proof of incomplete
> The term "complete" is nonsense. it implies that there are "incomplete" sets,

No, it describes what you believe. Must there be men with wings in order make the expression of men without wings meaningful? No. It is true that all men are without wings. That is not nonsense.

> sets that should contain something that they do not. All sets are "complete",

Then we can use this word without restriction.

> all sets contain exactly what they contain.

Nut we see here that they do not contain something completein the mapping.

> There are subsets and proper subsets but there is no such thing as an "incomplete" set.
>
> If a "proof" uses the word "complete" it is nonsense.

because it describes what you believe? It is what Cantor claims to exist and needs to "prove" equinumerosity. Is his statement nonsense?

"Wenn zwei wohldefinierte Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für Element, einander zuordnen lassen (was, wenn es auf eine Art möglich ist, immer auch noch auf viele andere Weisen geschehen kann), so möge für das Folgende die Ausdrucksweise gestattet sein, daß diese Mannigfaltigkeiten gleiche Mächtigkeit haben, oder auch, daß sie äquivalent sind." [Cantor, p. 119]

"gegenseitig eindeutige und vollständige Korrespondenz" [Cantor, p. 238 ]

"Die sämtlichen Punkte l unsrer Menge L sind also in gegenseitig eindeutige und vollständige Beziehung zu sämtlichen Punkten f der Menge F gebracht," [Cantor, p. 241]

"Zwei wohlgeordnete Mengen M und N heissen von gleichem Typus oder auch von gleicher Anzahl, wenn sie sich gegenseitig eindeutig und vollständig unter beidseitiger Wahrung der Rangfolge ihrer Elemente auf einander beziehen, abbilden lassen;" [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

Completed = vollständig

There was no objection to a 'potential infinity' in the form of an unending process, but an 'actual infinity' in the form of a completed infinite set was harder to accept." [H.B. Enderton: "Elements of set theory", Academic Press, New York (1977) p. 14f]

Regards, WM

On 8/22/2021 1:02 PM, WM wrote:
> William schrieb am Samstag, 21. August 2021 um 23:35:28 UTC+2:
>> On Saturday, August 21, 2021 at 5:28:38 PM UTC-4, WM wrote:
>
>>>> The reason I don't like "potential infinity" is that it is nonsense. It does not exist outside of Wolkenmuekenheim.
>>> There you are very wrong.
>>> "A potential infinity is a quantity which is finite but indefinitely large. For instance, when we enumerate the natural numbers as 0, 1, 2, ..., n, n+1, ..., the enumeration is finite at any point in time, but it grows indefinitely and without bound. [...] An actual infinity is a completed infinite totality.
>>>
>> The use of the word "completed" gives this away. It is nonsense. It changes nothing.
>
> Here is the proof of incomplete infinity:
>
> (1) When indexing the rationals all unit intervals can be completed.

red herring.

> (2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.

This simply shows your approach is wrong.

> (3) So there are at least two indices required to complete all intervals.

Wrong.

> (4) You cannot find out their order.

red herring.

>
> Although you prefer not to take notice of this fact, it is nevertheless unavoidable.

they are not facts at all, but your conjecture.
Without any context given, your above 4 lines are meaningless.

You post is trolling.

>
> Regards, WM
>

On 8/22/2021 1:14 PM, WM wrote:
> Jim Burns schrieb am Sonntag, 22. August 2021 um 00:24:41 UTC+2:
>> On 8/21/2021 5:32 PM, WM wrote:
>
>>> Then the completing is accomplished by belief only?
>> There are infinitely-many indexed rationals in the
>> interval (n,n+1], each rational with a finite index.
>
> Infinitely many is right, but completed infinity does not exist other than being dark.
> Proof:
>
> (1) When indexing the rationals all unit intervals can be completed.
> (2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
> (3) So there are at least two indices required to complete all intervals.
> (4) You cannot find out their order.

Fail. That is not a proof at all. the words "completed" "dark" "issued" are not math terms.
the sentences have no context and are unrelated.

On Sunday, August 22, 2021 at 2:46:49 PM UTC-4, WM wrote:
> > > Here is the proof of incomplete
> > The term "complete" is nonsense. it implies that there are "incomplete" sets,
> No

Yes. All sets contain what they contains. A set can be a proper subset but it cannot be "incomplete". If x is a FISON then x is an element of set set of FISONS.
The set of FISONs is not "incomplete" because the set has no last element.

--
William Hughes

On 8/22/2021 2:14 PM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 22. August 2021 um 00:24:41 UTC+2:
>> On 8/21/2021 5:32 PM, WM wrote:

>>> Then the completing is accomplished by belief only?
>> There are infinitely-many indexed rationals in the
>> interval (n,n+1], each rational with a finite index.
>
> Infinitely many is right, but completed infinity does not exist
> other than being dark.

We (broadly) have developed _variables_ by which we may refer
to _one of_ some collection without specifying _which one_

This is very useful for partially describing each individual in
the collection, even if there are infinitely-many individuals,
even if we ourselves are finite. The key here is to NOT SPECIFY
which individual in the collection is referred to and to only make
claims that are TRUE NO MATTER WHICH individual is referred to.

If we proceed from this partial description true of each
individual of interest using _only_ truth-preserving inferences,
we will be able to derive further claims that we will
know to be true of the same individuals, even if there are
infinitely-many individuals, even if we ourselves are finite.

Is this completed infinity? Are there dark numbers?
The answers to questions like these depend upon what you (WM)
unreliably mean by the terms.
What you (WM) unreliably mean by the terms does not affect
whether _variables_ et al. _work_
They work.

> Proof:
>
> (1) When indexing the rationals all unit intervals
> can be completed.

*No rational completes any interval*

For gcd(p,q) = 1, let
index(p/q) = q*q*p*p/rad(p)

For finite p, finite q, index(p/q) is finite.
All rationals are finitely indexed.

For each finitely-indexed rational r[j], there are
finitely-indexed rationals r[k] in (n,n+1] with k > j.

*No rational completes any interval*

I consider "there are more later" to imply "not completed".

> (2) Not all unit intervals can be completed by one index
> because an index is issued to one rational of one interval only.
> (3) So there are at least two indices required to complete
> all intervals.
> (4) You cannot find out their order.
>
> Although you prefer not to take notice of this fact,
> it is nevertheless unavoidable.

*No rational completes any interval*

>> ( Use index(p/q) = q*q*p*p/rad(p) with gcd(p,q) = 1
>> (
>> ( Assume that there are finitely-many indexed rationals.
>
> That is not the alternative.
>
>> There is no natural-number/finite-index after infinitely-many
>> indexed rationals, each with a finite index.
>
> Correct. There is potential infinity. Not possible to complete it.
> This is proved above.

Is this completed infinity? Are there dark numbers?
The answers to questions like these depend upon what you (WM)
unreliably mean by the terms.
What you (WM) unreliably mean by the terms does not affect
whether _variables_ et al. _work_
They work.

> Look: The mapping of |N can show that there are as many rational
> numbers in the interval (0, 10^-100000) as in (-oo, oo), as many
> prime numbers as algebraic numbers or definable real and complex
> numbers. Don't you suspect an error of the counter-intuitive
> interpretation?

What makes our _uneducated_ intuition better than our _educated_
intuition?

Remind me. Why are we discussing any of this, if we've ruled out
changing one's mind from the get-go?

> The explanation is very simple and will convince every student
> not yet infected too much by set theory:

All right triangles are right triangles.
All natural numbers are natural numbers.

We can describe what we mean by "right triangle", by "natural
number", and then reason truth-preserving-ly from the description.
We derive further claims about right triangles or about natural
numbers that we know are true independently of any potential
infinities or actual infinities involved.
Natural numbers are natural numbers.
Preserving truth preserves truth.
Done.

> There is a potentially infinite set of definable elements
> in every infinite set. They can be mapped to each other.
> That's why all sets have same cardinality.

The set of definable elements contains only definable elements.
You see a "paradox" in that.

It's not any source of concern for us.
Natural numbers are natural numbers.
Preserving truth preserves truth.
Done.

Jim Burns schrieb am Sonntag, 22. August 2021 um 21:45:36 UTC+2:

> *No rational completes any interval*

Then there is no bijection and no cardinality. For cardinality completion is required.

There was no objection to a 'potential infinity' in the form of an unending process, but an 'actual infinity' in the form of a completed infinite set was harder to accept." [H.B. Enderton: "Elements of set theory", Academic Press, New York (1977) p. 14f]

Regards, WM

William schrieb am Sonntag, 22. August 2021 um 21:04:03 UTC+2:
> On Sunday, August 22, 2021 at 2:46:49 PM UTC-4, WM wrote:
> > > > Here is the proof of incomplete
> > > The term "complete" is nonsense. it implies that there are "incomplete" sets,
> > No
> Yes. All sets contain what they contains. A set can be a proper subset but it cannot be "incomplete". If x is a FISON then x is an element of set set of FISONS.
> The set of FISONs is not "incomplete" because the set has no last element.
>
But a mapping can be incomplete. As I have shown it is incomplete. But according to Cantor, it should be complete:

There was no objection to a 'potential infinity' in the form of an unending process, but an 'actual infinity' in the form of a completed infinite set was harder to accept." [H.B. Enderton: "Elements of set theory", Academic Press, New York (1977) p. 14f]

Regards, WM

On Sunday, 22 August 2021 at 19:00:49 UTC-3, WM wrote:
> Jim Burns schrieb am Sonntag, 22. August 2021 um 21:45:36 UTC+2:
>
>
> > *No rational completes any interval*
> Then there is no bijection and no cardinality. For cardinality completion is required.

If you cannot understand that there is no largest natural number, how can you *EVER* even *THINK* you might have a single fucking clue about rational numbers and Cantor's counting process?

On Sunday, August 22, 2021 at 6:02:18 PM UTC-4, WM wrote:
> William schrieb am Sonntag, 22. August 2021 um 21:04:03 UTC+2:
> > On Sunday, August 22, 2021 at 2:46:49 PM UTC-4, WM wrote:
> > > > > Here is the proof of incomplete
> > > > The term "complete" is nonsense. it implies that there are "incomplete" sets,
> > > No
> > Yes. All sets contain what they contains. A set can be a proper subset but it cannot be "incomplete". If x is a FISON then x is an element of set set of FISONS.
> > The set of FISONs is not "incomplete" because the set has no last element.
> >
> But a mapping can be incomplete.

Nope, a bijection is a set of ordered pairs. There is no such thing as an "incomplete" set. The set of FISONs has no last element but there is no FISON missing from the set.

--
Willaim Hughes

torsdag 19 augusti 2021 kl. 14:04:24 UTC+2 skrev WM:
> Sergio schrieb am Mittwoch, 18. August 2021 um 23:32:47 UTC+2:
>
> > Your admission that Dark Numbers are a Belief, and not fact does help see your point of view.
> Dark numbers are only existing if infinity can be completed. Then they are necessary. In classical mathematics they are useless.
>
> >> But it is clear that not all can be completed by one index.
>
> > wrong.
>
> Every index is issued into one unit interval. Therefore it is impossible that by an index issued to interval n other intervals can be completed.
> (1) You claim that all intervals can be completed.
> (2) Not all intervals can be completed by one index.
> (3) So there are at least two indices required to complete all intervals.
> (4) You cannot find out their order by their FISONs.
> (5) But you claim that only indices having FISONs are existing.
> Regards, WM
your "dark numbers" have no meaning, they do not exist cause you cannot meaningfully define them.

torsdag 19 augusti 2021 kl. 14:19:46 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 19. August 2021 um 07:00:48 UTC+2:
> > >The definable natural numbers have a discernible well-ordering
> > Whatever "discernible" means.
> It can be verified step by step.
> >
> > but N is well-ordered so again, N_def=N
> >
> No:
> (1) When indexing the rationals all unit intervals can be completed.
> (2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
> (3) So there are at least two indices required to complete all intervals.
> (4) You cannot find out their order by their FISONs. If you could, you would complete aleph_0 rationals of one interval but you would need further indices. Contradiction. That is the true reason for dark numbers.
>
> Regards, WM
mathematics doesn't work in "steps" like a fucking program.

N is well-ordered and your N_def=N

On 8/22/2021 6:00 PM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 22. August 2021 um 21:45:36 UTC+2:

>> *No rational completes any interval*
>
> Then there is no bijection and no cardinality.
> For cardinality completion is required.
>
> There was no objection to a 'potential infinity' in the
> form of an unending process, but an 'actual infinity' in
> the form of a completed infinite set was harder to accept."
> [H.B. Enderton: "Elements of set theory",
> Academic Press, New York (1977) p. 14f]

"p. 14f" would be the forward. Is that right?

----
Should I assume from your use of that quote that you do not
accept the assumption that completed infinite sets exist?

Perhaps you accept different assumptions.
| A finite index is a finite index.
| A positive rational is a positive rational.
| Preserving truth preserves truth.

Do you (WM) accept these assumptions?