William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
> On Thursday, August 19, 2021 at 1:53:53 PM UTC-4, WM wrote:
> > William schrieb am Donnerstag, 19. August 2021 um 19:24:31 UTC+2:
> > >
> you cannot use a step by step method (something you call a "well ordered way") to talk about the "completetion" of a set with no last element.
> > You cannot use it at all for infinite sets.
> Use other methods. Get results.

The methods you use are based upon bijections, i.e., one-to-one mappings provable step-by-step. You have seen that they do no longer hold (they never did) but you claim that the dark action is sufficient to prove mappings. It is not.

> Note that a contradiction is not a result that you think is "Repugnant to the nature of mathematics"

Whatever, it is, your "new methods" are required by dark numbers only. Here is the 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 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

On Thursday, August 19, 2021 at 7:24:19 PM UTC+2, WM wrote:

> It is hard to imagine any other reason for including [zero] in the set of natural numbers.

Only for idiots like you.

Hint: What is the number (germ. Anzahl) of gold coins in your pockets?

On Thursday, August 19, 2021 at 7:53:53 PM UTC+2, WM wrote:

> Cantor's bijection [...] assumes that step-by-step is always possible and <bla>

No, "it" doesn't, you silly crank.

On Thursday, August 19, 2021 at 7:24:31 PM UTC+2, William wrote:
> On Thursday, August 19, 2021 at 1:13:51 PM UTC-4, WM wrote:
> >
> > I only said that if |N is a sequence and <bla>
> >
> Correct

NO, this is NOT correct. IN ISN'T a sequence.

Hint: {1, 2, 3, ...} =/= (1, 2, 3, ...)

HOLY SHIT!!!

More hints:

1. (1, 2, 3, ...) = {(1,1), (2,2), (3,3), ...}.

2. IN \ {1, 2, 3, ...} = {}, but IN \ (1, 2, 3, ...) = IN.

On Thursday, August 19, 2021 at 8:35:52 PM UTC+2, WM wrote:
> William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2: [something]
>
> you claim that the dark action is sufficient to prove mappings.

Where did William claim that? Can you present a quote please?

On Thursday, August 19, 2021 at 5:45:07 PM UTC+2, Python wrote: [...]

> Only Crank Mueckenheim, unfortunately, is teaching at Hochschule Augsburg.

Yes, it's a shame!

On 8/18/2021 5:06 PM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 18. August 2021 um 22:50:40 UTC+2:

>> That would be all the rationals, since all rationals
>> are finitely-indexed. (See q*q*p*p/rad(p).)
>> There is no further indexing left to be done.
>
> So you have indexed at least one rational in aleph_0 intervals
> and have used aleph_0 natural numbers.
> What do you use for completing the intervals?

If aleph_0 rationals have been indexed, then each rational
_with a finite index_ has been indexed.

Using index(p/q) = q*q*p*p/rad(p), each positive rational
has a finite index.

If aleph_0 rationals have been indexed, then each rational
has been indexed.

There's nothing that needs completing.

>>>> To be clear, these rationals are definable positive rationals
>>>> and these indexes are definable finite indexes.
>>>
>>> Then define the indexes completing the intervals - whatever you
>>> understand by completing. But don't try to claim that all
>>> incomplete intervals are completed by one index.
>
>> What I understand is that we cannot, even in principle,
>> step through an infinite set one-by-one.
>
> Above you said that all indexes are finite.
> What finite index cannot be stepped to? And why?

Each finite index can be stepped to,
ignoring physical constraints.
For each k, steppable 0,...,k with adjacent i,j = i+1 exists.

There is no finite index k such that, upon stepping to k,
all of the set of finite indexes has been stepped to.
For each steppable 0,...,k with adjacent i,j = i+1,
k+1 is not in 0,...,k.

WM formulated on Thursday :
> Gus Gassmann schrieb am Donnerstag, 19. August 2021 um 17:41:52 UTC+2:
>> On Thursday, 19 August 2021 at 12:20:55 UTC-3, Greg Cunt wrote:
>>> On Thursday, August 19, 2021 at 2:12:01 PM UTC+2, WM wrote:
>>>> Greg Cunt schrieb am Mittwoch, 18. August 2021 um 23:42:00 UTC+2:
>>>>
>>>> "In der DIN-Norm 5473 wird die Null zu den natürlichen Zahlen gezählt."
>>>> ["In DIN standard 5473, zero is counted as a natural number."]
>>>>
>>>> Written by matheologians who try to veil the mistakes of set theory.
>>> Oh, I see, there's a conspiracy going on. The Deutsches Institut für
>>> Normung has been infiltrated by matheologians who try to veil the mistakes
>>> of set theory.
>
> Obviously. It is hard to imagine any other reason for including one of the
> most unnatural numbers in the set of natural numbers.

It provides structure.

On 8/19/2021 2:30 PM, Greg Cunt wrote:
> On Thursday, August 19, 2021 at 5:45:07 PM UTC+2, Python wrote: [...]
>
>> Only Crank Mueckenheim, unfortunately, is teaching at Hochschule Augsburg.
>
> Yes, it's a shame!
>

if his students have any math, they will know he is a crank

On 8/19/2021 2:25 PM, Greg Cunt wrote:
> On Thursday, August 19, 2021 at 8:35:52 PM UTC+2, WM wrote:
>> William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2: [something]
>>
> > you claim that the dark action is sufficient to prove mappings.
>
> Where did William claim that? Can you present a quote please?
>

there are dark maps.

(insert more bs here)

Ants with Dark Maps

On Thursday, August 19, 2021 at 2:35:52 PM UTC-4, WM wrote:
> William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
> > On Thursday, August 19, 2021 at 1:53:53 PM UTC-4, WM wrote:
> > > William schrieb am Donnerstag, 19. August 2021 um 19:24:31 UTC+2:
> > > >
> > you cannot use a step by step method (something you call a "well ordered way") to talk about the "completetion" of a set with no last element.
> > > You cannot use it at all for infinite sets.
> > Use other methods. Get results.
> The methods you use are based upon bijections,

Nope, Non step by step methods lead to bijections. They are not based on bijections. They lead to results that you think are "Repugnant to the nature of mathematics" not to contradictions.

--
William Hughes

Sergio wrote:

> On 8/19/2021 2:25 PM, Greg Cunt wrote:
>> On Thursday, August 19, 2021 at 8:35:52 PM UTC+2, WM wrote:
>>> William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
>>> [something]
>>>
>> > you claim that the dark action is sufficient to prove mappings.
>>
>> Where did William claim that? Can you present a quote please?
>>
>>
> there are dark maps.

pull my finger.

On 8/19/2021 3:50 PM, Cash Bornhoft wrote:
> Sergio wrote:
>
>> On 8/19/2021 2:25 PM, Greg Cunt wrote:
>>> On Thursday, August 19, 2021 at 8:35:52 PM UTC+2, WM wrote:
>>>> William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
>>>> [something]
>>>>
>>> > you claim that the dark action is sufficient to prove mappings.
>>>
>>> Where did William claim that? Can you present a quote please?
>>>
>>>
>> there are dark maps.
>
> pull my finger.
>

No! it is dark finger!

William schrieb am Donnerstag, 19. August 2021 um 22:48:48 UTC+2:
> On Thursday, August 19, 2021 at 2:35:52 PM UTC-4, WM wrote:
> > William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
> > > On Thursday, August 19, 2021 at 1:53:53 PM UTC-4, WM wrote:
> > > > William schrieb am Donnerstag, 19. August 2021 um 19:24:31 UTC+2:
> > > > >
> > > you cannot use a step by step method (something you call a "well ordered way") to talk about the "completetion" of a set with no last element.
> > > > You cannot use it at all for infinite sets.
> > > Use other methods. Get results.
> > The methods you use are based upon bijections,
> Nope, Non step by step methods lead to bijections.

For every FISON a bijection with |N can be analyzed step by step. If this ceases then there are no FISONs involved.

> They are not based on bijections. They lead to results that you think are "Repugnant to the nature of mathematics" not to contradictions.

You claim that all rationals in all intervals can be indexed completely. You claim that all indexes can be put in well-order. You know that not two or more intervals can be completed by one index. Therefore at least two indexes are necessary. But you don't know their order. Therefore your claim is null and void.

Do you withdraw your claim that all natural can be put in a well-order?

Regards, WM

Greg Cunt schrieb am Donnerstag, 19. August 2021 um 21:25:48 UTC+2:
> On Thursday, August 19, 2021 at 8:35:52 PM UTC+2, WM wrote:
> > William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2: [something]
> >
> > you claim that the dark action is sufficient to prove mappings.
> Where did William claim that? Can you present a quote please?

"Non step by step methods lead to bijections." (WH) "Non step by step" is dark because "not dark" means step by step is possible. Of course it is nonsense to claim that they lead to beiections.

Regards, WM

Greg Cunt schrieb am Donnerstag, 19. August 2021 um 20:55:36 UTC+2:
> On Thursday, August 19, 2021 at 7:24:19 PM UTC+2, WM wrote:
>
> > It is hard to imagine any other reason for including [zero] in the set of natural numbers.
>
> Hint: What is the number (germ. Anzahl) of gold coins in your pockets?

There is none or zero, therefore there is no natural number of coins.

Regards, WM

On Friday, August 20, 2021 at 2:16:04 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Donnerstag, 19. August 2021 um 20:55:36 UTC+2:
> > On Thursday, August 19, 2021 at 7:24:19 PM UTC+2, WM wrote:
> > >
> > > It is hard to imagine any other reason for including [zero] in the set of natural numbers.
> > >
> > Hint: What is the number (germ. Anzahl) of gold coins in your pockets?
> >
> There is none or <bla>

That's not an answer to my question. I asked: "What is the number (germ. Anzahl) of gold coins in your pockets?" I didn't ask "Is there a gold coins in your pockets?" - Got it, dumbo?

Actually, the number of gold coins in your pockets is

> zero

Indeed! :-)

That's why 0 is considered a natural number. A natural number gives the answer to the question "How many?".

On Friday, August 20, 2021 at 2:14:08 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Donnerstag, 19. August 2021 um 21:25:48 UTC+2:
> > On Thursday, August 19, 2021 at 8:35:52 PM UTC+2, WM wrote:
> > > William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2: [something]
> > >
> > > you claim that the dark action is sufficient to prove mappings.
> > >
> > Where did William claim that? Can you present a quote please?
> >
> "Non step by step methods lead to bijections." (WH)

Huh?! I asked for a quote that supports your claim that William claimed that "the dark action is sufficient to prove mappings".

Hint: The quote you presented does not even contain the word "dark". Moreover it does NOT state that some "action is sufficient to prove mappings" --- HOLY SHIT!

You are a fraud, WM!

On 8/20/2021 7:09 AM, WM wrote:
> William schrieb am Donnerstag, 19. August 2021 um 22:48:48 UTC+2:
>> On Thursday, August 19, 2021 at 2:35:52 PM UTC-4, WM wrote:
>>> William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
>>>> On Thursday, August 19, 2021 at 1:53:53 PM UTC-4, WM wrote:
>>>>> William schrieb am Donnerstag, 19. August 2021 um 19:24:31 UTC+2:
>>>>>>
>>>> you cannot use a step by step method (something you call a "well ordered way") to talk about the "completetion" of a set with no last element.
>>>>> You cannot use it at all for infinite sets.
>>>> Use other methods. Get results.
>>> The methods you use are based upon bijections,
>> Nope, Non step by step methods lead to bijections.
>
> For every FISON a bijection with |N can be analyzed step by step. If this ceases then there are no FISONs involved.

also this is true;
For every AAAAA a bijection with |N can be analyzed step by step. If this ceases then there are no AAAAA involved.

>
>> They are not based on bijections. They lead to results that you think are "Repugnant to the nature of mathematics" not to contradictions.
>
> You claim that all rationals in all intervals can be indexed completely.

obviously true, simply remove interval boundries, all rationals are indexed with one index, the real numbers.

this shows that your 'intervals" are distraction.

> You claim that all indexes can be put in well-order.

can the natural numbers be put in well-order ?

> You know that not two or more intervals can be completed by one index.

those are your thoughts, not his. and it is wrong. another index is not needed at all.

simply remove interval boundries, all rationals are indexed with one index, the real numbers.

> Therefore at least two indexes are necessary.

Wrong.

> But you don't know their order.

also wrong as two indexes are not needed.

> Therefore your claim is null and void.

wrong, you make several mistakes

>
> Do you withdraw your claim that all natural can be put in a well-order?

The standard ordering ≤ of the natural numbers is a well ordering

>
> Regards, WM
>

Jim Burns schrieb am Donnerstag, 19. August 2021 um 21:58:57 UTC+2:
> On 8/18/2021 5:06 PM, WM wrote:

> > So you have indexed at least one rational in aleph_0 intervals
> > and have used aleph_0 natural numbers.
> > What do you use for completing the intervals?
>
> If aleph_0 rationals have been indexed, then each rational
> _with a finite index_ has been indexed.

That is at least the caee when the first interval has been completed. But what happens then?
>
> If aleph_0 rationals have been indexed, then each rational
> has been indexed.

Wrong. aleph_0 rationals are in the first completed interval.
>
> There's nothing that needs completing.

No? It is completed in the delusions of matheologians without any indexing.
>
> >> What I understand is that we cannot, even in principle,
> >> step through an infinite set one-by-one.
> >
> > Above you said that all indexes are finite.
> > What finite index cannot be stepped to? And why?
>
> Each finite index can be stepped to,
> ignoring physical constraints.

Why then can't you find out which interval is completed first?
> For each k, steppable 0,...,k with adjacent i,j = i+1 exists.
>
> There is no finite index k such that, upon stepping to k,
> all of the set of finite indexes has been stepped to.

But othere indices do not exist in order to accomplish what you claim to believe: "If two well-defined manifolds, M and N, can be related completely, element by element, to each other [...], then for the following the expression may be permitted that these manifolds have the same cardinality or that they are equivalent." [Cantor]

> For each steppable 0,...,k with adjacent i,j = i+1,
> k+1 is not in 0,...,k.

Therefore your belief described above is nonsense. Every definable k is not sufficient. And undefinable elements cannot be mapped. If you are not irreversibly brain damaged, like most matheologians, then you will recognize this.

Regards, WM

Greg Cunt schrieb am Freitag, 20. August 2021 um 14:55:57 UTC+2:
> On Friday, August 20, 2021 at 2:14:08 PM UTC+2, WM wrote:
> > Greg Cunt schrieb am Donnerstag, 19. August 2021 um 21:25:48 UTC+2:
> > > On Thursday, August 19, 2021 at 8:35:52 PM UTC+2, WM wrote:
> > > > William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2: [something]
> > > >
> > > > you claim that the dark action is sufficient to prove mappings.
> > > >
> > > Where did William claim that? Can you present a quote please?
> > >
> > "Non step by step methods lead to bijections." (WH)
> Huh?! I asked for a quote that supports your claim that William claimed that "the dark action is sufficient to prove mappings".

You need the quote quoted above by me and some brain capacity that unfortunately cannot be provided by me.

Regards, WM

On 8/20/2021 9:46 AM, WM wrote:
> Jim Burns schrieb am Donnerstag, 19. August 2021 um 21:58:57 UTC+2:
>> On 8/18/2021 5:06 PM, WM wrote:
>
>>> So you have indexed at least one rational in aleph_0 intervals
>>> and have used aleph_0 natural numbers.
>>> What do you use for completing the intervals?
>>
>> If aleph_0 rationals have been indexed, then each rational
>> _with a finite index_ has been indexed.
>
> That is at least the caee when the first interval has been completed. But what happens then?

the first interval has been cut into 1/3's, and is now 3 sub intervals.

WM now requires 3 more indexes to be used. Why ?

>>
>> If aleph_0 rationals have been indexed, then each rational
>> has been indexed.
>
> Wrong. aleph_0 rationals are in the first completed interval.

Wrong. you are thinking of finite numbers.

>>
>> There's nothing that needs completing.
>
> No? It is completed in the delusions of matheologians without any indexing.
>>
>>>> What I understand is that we cannot, even in principle,
>>>> step through an infinite set one-by-one.
>>>
>>> Above you said that all indexes are finite.
>>> What finite index cannot be stepped to? And why?
>>
>> Each finite index can be stepped to,
>> ignoring physical constraints.
>
> Why then can't you find out which interval is completed first?

because they are infinite.

>
>> For each k, steppable 0,...,k with adjacent i,j = i+1 exists.
>>
>> There is no finite index k such that, upon stepping to k,
>> all of the set of finite indexes has been stepped to.
>
> But othere indices do not exist in order to accomplish what you claim to believe: "If two well-defined manifolds, M and N, can be related completely, element by element, to each other [...], then for the following the expression may be permitted that these manifolds have the same cardinality or that they are equivalent." [Cantor]
>
>> For each steppable 0,...,k with adjacent i,j = i+1,
>> k+1 is not in 0,...,k.
>
> Therefore your belief described above is nonsense. Every definable k is not sufficient. And undefinable elements cannot be mapped. If you are not irreversibly brain damaged, like most matheologians, then you will recognize this.

Calling others names that find your mistakes for you is act of desperation, as you have no good arguments left to support what you present.

>
> Regards, WM
>

On Friday, August 20, 2021 at 4:50:09 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Freitag, 20. August 2021 um 14:55:57 UTC+2:
> > On Friday, August 20, 2021 at 2:14:08 PM UTC+2, WM wrote:
> > > Greg Cunt schrieb am Donnerstag, 19. August 2021 um 21:25:48 UTC+2:
> > > > On Thursday, August 19, 2021 at 8:35:52 PM UTC+2, WM wrote:
> > > > > William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2: [something]
> > > > >
> > > > > you claim that the dark action is sufficient to prove mappings.
> > > > >
> > > > Where did William claim that? Can you present a quote please?
> > > >
> > > "Non step by step methods lead to bijections." (WH)
> > >
> > Huh?! I asked for a quote that supports your claim that William claimed that "the dark action is sufficient to prove mappings".
> >
> [...] unfortunately cannot be provided by me.

I see. So you are fraud AND a liar, WM, thought so.

On Friday, August 20, 2021 at 8:09:53 AM UTC-4, WM wrote:
> William schrieb am Donnerstag, 19. August 2021 um 22:48:48 UTC+2:
> > On Thursday, August 19, 2021 at 2:35:52 PM UTC-4, WM wrote:
> > > William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
> > > > On Thursday, August 19, 2021 at 1:53:53 PM UTC-4, WM wrote:
> > > > > William schrieb am Donnerstag, 19. August 2021 um 19:24:31 UTC+2:
> > > > > >
> > > > you cannot use a step by step method (something you call a "well ordered way") to talk about the "completetion" of a set with no last element.
> > > > > You cannot use it at all for infinite sets.
> > > > Use other methods. Get results.
> > > The methods you use are based upon bijections,
> > Nope, Non step by step methods lead to bijections.
> For every [element of the set of FISONs ] a bijection with |N can be analyzed step by step

Look! Over there! A Pink Elephant!
The bijection between |N and the *set of FISONs* can be analyzed step by step.

--
William huges

On 8/20/2021 10:46 AM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 19. August 2021 um 21:58:57 UTC+2:
>> On 8/18/2021 5:06 PM, WM wrote:

>>> So you have indexed at least one rational in aleph_0
>>> intervals and have used aleph_0 natural numbers.
>>> What do you use for completing the intervals?
>>
>> If aleph_0 rationals have been indexed, then each rational
>> _with a finite index_ has been indexed.
>
> That is at least the caee when the first interval has been
> completed. But what happens then?

Each rational has a finite index.
Since each rational with a finite index has been indexed,
each rational has been indexed.

>> If aleph_0 rationals have been indexed, then each rational
>> has been indexed.
>
> Wrong. aleph_0 rationals are in the first completed interval.

The first completed interval [k,k+1] -- if it existed --
would have aleph_0 rationals indexed while all other intervals
still had one or more _finitely-indexed_ rationals not yet
indexed. That's a contradictory description.
First completed [k,k+1] does not exist.

>> There's nothing that needs completing.
>
> No? It is completed in the delusions of matheologians
> without any indexing.

We can make claims about one of infinitely-many individuals
true of that individual no matter which individual is referred to.
( "One angle of that right triangle is a right angle." )

We can find further claims by truth-preserving inferences,
starting from those initial, descriptive claims.
Because truth-preserving, the further claims will also be
true of each one of the infinitely-many individuals the
initial claims are true of.

Another way to say this is that the further claims are
true of the _complete collection_ of individuals described.
That seems to be what you're calling a delusion.
If it's a delusion, then it is a delusion that
a right triangle is a right triangle, or that a natural
number is natural number.

Your (WM's) argument, so-called, after all the goo and
dribble has been wrung out of it, leaves a blank place
on the screen. It contains nothing other than goo and dribble.

>>>> What I understand is that we cannot, even in principle,
>>>> step through an infinite set one-by-one.
>>>
>>> Above you said that all indexes are finite.
>>> What finite index cannot be stepped to? And why?
>>
>> Each finite index can be stepped to,
>> ignoring physical constraints.
>
> Why then can't you find out which interval is completed first?

The first completed interval [k,k+1] -- if it existed --
would have aleph_0 rationals indexed while all other intervals
still had one or more _finitely-indexed_ rationals not yet
indexed. That's a contradictory description.
First completed [k,k+1] does not exist.

>> For each k, steppable 0,...,k with adjacent i,j = i+1 exists.
>>
>> There is no finite index k such that, upon stepping to k,
>> all of the set of finite indexes has been stepped to.
>
> But othere indices do not exist in order to accomplish what
> you claim to believe:

For each steppable 0,...,k with adjacent i,j = i+1,
k+1 is not in 0,...,k.

Are these the indices which supposedly do not exist?

> "If two well-defined manifolds, M and N, can be related
> completely, element by element, to each other [...], then
> for the following the expression may be permitted that
> these manifolds have the same cardinality or that they
> are equivalent." [Cantor]
>
>> For each steppable 0,...,k with adjacent i,j = i+1,
>> k+1 is not in 0,...,k.
>
> Therefore your belief described above is nonsense.
> Every definable k is not sufficient.
> And undefinable elements cannot be mapped.
> If you are not irreversibly brain damaged,
> like most matheologians, then you will recognize this.

No finite index is preceded by infinitely-many indexes.
Agreed.
Other than that, goo and dribble.