Greg Cunt formulated the question :
> On Wednesday, August 18, 2021 at 10:07:31 PM UTC+2, FromTheRafters wrote:
>> WM explained :
>>> FromTheRafters schrieb am Mittwoch, 18. August 2021 um 16:28:42 UTC+2:
>>>> WM formulated:
>>>>>
>>>>> N is the limit of the sequence of FISONs.
>>>>>
>>>> The sequence of FISONs has a limit?
>>>>
>>> Yes, like the sequence (1/n) has the limit 0.
>
> Ah?

It converges in the reals. This 'notion' is not in the 'set' realm, but
WM likes to blur the distinctions. When the sequence or associated
series are not convergent, they are called divergent and there is no
'limit'.

>> So, the sequence of FISONs [...] converges to a number in the set of natural
>> numbers?
>
> C'mon, that's not fair. We all know that WM is an idiot. But h e r e
> there's nothing (!) to complain, I's say.
>
> 1. "N is the limit of the sequence of FISONs." (true)

There is a difference between when you state it and when WM states it.
Here you state it and it remains in context with "Set Theoretical
Limit" and with WM it is a trick. I try to get him to commit to
something.

> 2. "The sequence (1/n) has the limit 0." (true)
>
> 3. "... like ..." (true, in a certain sense. After all, a set theoretic limit
> is a "limit" - sort of. There are formal similarities between the notion of
> /set-theoretic limit/ and the "limit" defined for sequences of real numbers).
>
> See: https://en.wikipedia.org/wiki/Set-theoretic_limit

Yes, and one can equal zero in a certain context too. I was only trying
to ask WM how STL applies to sequences of strictly monotonic increases
in cardinality. Basically, these FISONs cardinalities are just the
natural numbers wearing party hats. -- their related sequence of
partial sums are divergent as is the sum of all naturals.

Yes, I am aware of the -1/12 figure from the double inversion but lets
not bring WM there, he is already having enough trouble seeing numbers
as they keep going dark on him.

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

Jim Burns schrieb am Mittwoch, 18. August 2021 um 23:38:55 UTC+2:

> No finite index can complete an interval with infinitely-many
> indexed rationals.

How then can it be completed?

Regards, WM

Greg Cunt schrieb am Mittwoch, 18. August 2021 um 23:42:00 UTC+2:
> On Wednesday, August 18, 2021 at 10:37:55 PM UTC+2, WM wrote:
>
> > 0 is not a natural number
> Really?
>
> Hint: "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.

Why zero has been included in the set of natural numbers

For every natural number we have

|{1, 2, 3, ..., n}| = n

and, if a proper limit X exists,

lim_{n-->X} |{1, 2, 3, ..., n}| = lim_{n-->X} n .

Hence we have always

|{1, 2, 3, ..., X}| = X .

The following equations suggest themselves

|{1, 2, 3, ..., ℵo}| = ℵo
(1, 2, 3, ..., ω) = ω

but are unpopular in set theory. Correct is there

(1, 2, 3, ..., ω) = ω + 1 .

This is much easier to learn (and to teach) if the pupil starts with

|{0, 1, 2, 3, ..., n-1}| = n .

Regards, WM

William schrieb am Donnerstag, 19. August 2021 um 01:13:44 UTC+2:
> On Wednesday, August 18, 2021 at 3:38:35 PM UTC-4, WM wrote:
> > William schrieb am Mittwoch, 18. August 2021 um 18:32:37 UTC+2:
> > > On Wednesday, August 18, 2021 at 12:23:09 PM UTC-4, WM wrote:
> > > > William schrieb am Mittwoch, 18. August 2021 um 18:13:59 UTC+2:
> > > > > On Wednesday, August 18, 2021 at 7:43:08 AM UTC-4, WM wrote:
> > > >
> > > > > > Why not? For every definable index it has meaning. If it has no meaning later, then there are no definable indices.
> > > > > Nope, The fact that it has meaning for every index you can write down, does not mean it has meaning for every index you cannot write down.
> > > > Correct. It has no meaning for dark indices.
> > > The set |N_F does not contain "dark" elements.
> > How do you explain that many indexes can be issued in order at the beginning while no index can be distinguished after all intervals have been defiled?
> Nothing to explain as there is no problem in distinguishing elements of |N_F that cannot be written down.

They cannot be distinguished because they cannot be written down? No that's not the true reason. If they had a discernible well-order, then this would cause contradictions.

(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

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

on 8/18/2021, Sergio supposed :
> On 8/18/2021 2:38 PM, WM wrote:
>> William schrieb am Mittwoch, 18. August 2021 um 18:32:37 UTC+2:
>>> On Wednesday, August 18, 2021 at 12:23:09 PM UTC-4, WM wrote:
>>>> William schrieb am Mittwoch, 18. August 2021 um 18:13:59 UTC+2:
>>>>> On Wednesday, August 18, 2021 at 7:43:08 AM UTC-4, WM wrote:
>>>>>> Why not? For every definable index it has meaning. If it has no meaning
>>>>>> later, then there are no definable indices.
>>>>> Nope, The fact that it has meaning for every index you can write down,
>>>>> does not mean it has meaning for every index you cannot write down.
>>>> Correct. It has no meaning for dark indices.
>>> The set |N_F does not contain "dark" elements.
>>
>> How do you explain that many indexes can be issued in order at the beginning
>> while no index can be distinguished after all intervals have been defiled?
>
> in what context ?
>
>>>
>>>>> The set |N_F does not contain "dark" elements.
>>>> But if it indexes all fraction, then it contains elements that cannot be
>>>> ordered.
>>> Nope there is no element of U_F that cannot be indexed by |N_F and no
>>> element of |N_F that cannot be ordered.
>> How do you explain that no index can be ordered after all intervals have
>> been defiled?
>
> in what context ?

Defiled? It *is* almost criminal the way he defiles those little
intervals.

On Thursday, August 19, 2021 at 2:12:01 PM UTC+2, WM wrote:

> The following equations suggest themselves
>
> |{1, 2, 3, ..., ℵo}| = ℵo

Clearly true.

Hint: |{1, 2, 3, ...}| = ℵo. |{1, 2, 3, ..., ℵo}| = |{1, 2, 3, ...} u {ℵo}| = |{1, 2, 3, ...}| + |{ℵo}| = ℵo + 1 = ℵo (since ℵo !e {1, 2, 3, ...}).

> (1, 2, 3, ..., ω) = ω

Clearly nonsense.
> but are unpopular in set theory. Correct is there
>
> (1, 2, 3, ..., ω) = ω + 1 .

No, it isn't.

On 8/19/2021 7:11 AM, WM wrote:
> Greg Cunt schrieb am Mittwoch, 18. August 2021 um 23:42:00 UTC+2:
>> On Wednesday, August 18, 2021 at 10:37:55 PM UTC+2, WM wrote:
>>
>>> 0 is not a natural number
>> Really?
>>
>> Hint: "In der DIN-Norm 5473 wird die Null zu den natürlichen Zahlen gezählt." ["In DIN standard 5473, zero is counted as a natural number."]
>

<snip crap>

On Thursday, 19 August 2021 at 09:35:44 UTC-3, FromTheRafters wrote:
> on 8/18/2021, Sergio supposed :
> > On 8/18/2021 2:38 PM, WM wrote:
> >> William schrieb am Mittwoch, 18. August 2021 um 18:32:37 UTC+2:
> >>> On Wednesday, August 18, 2021 at 12:23:09 PM UTC-4, WM wrote:
> >>>> William schrieb am Mittwoch, 18. August 2021 um 18:13:59 UTC+2:
> >>>>> On Wednesday, August 18, 2021 at 7:43:08 AM UTC-4, WM wrote:
> >>>>>> Why not? For every definable index it has meaning. If it has no meaning
> >>>>>> later, then there are no definable indices.
> >>>>> Nope, The fact that it has meaning for every index you can write down,
> >>>>> does not mean it has meaning for every index you cannot write down.
> >>>> Correct. It has no meaning for dark indices.
> >>> The set |N_F does not contain "dark" elements.
> >>
> >> How do you explain that many indexes can be issued in order at the beginning
> >> while no index can be distinguished after all intervals have been defiled?
> >
> > in what context ?
> >
> >>>
> >>>>> The set |N_F does not contain "dark" elements.
> >>>> But if it indexes all fraction, then it contains elements that cannot be
> >>>> ordered.
> >>> Nope there is no element of U_F that cannot be indexed by |N_F and no
> >>> element of |N_F that cannot be ordered.
> >> How do you explain that no index can be ordered after all intervals have
> >> been defiled?
> >
> > in what context ?
> Defiled? It *is* almost criminal the way he defiles those little
> intervals.

Ah, the phantasies of a lapsed physicist!

On 8/19/2021 7:19 AM, WM wrote:
> 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.

by completed you mean finished ?

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

wrong.

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

wrong.

> (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.

wrong.

>
> Regards, WM
>

Score 3/4 wrong you get 25% fail

Return of the Interval Ants

On 8/19/2021 6:56 AM, WM wrote:
> Jim Burns schrieb am Mittwoch, 18. August 2021 um 23:23:07 UTC+2:
>> On 8/18/2021 3:49 PM, WM wrote:
>
>>> You say all can be completed.
>>> You know you cannot find out what is completed first.
>> I know that none of the intervals is complete before any of
>> the other intervals.
>
> (1) You claim that all intervals can be completed.

completed means finished ?

> (2) You know that not all intervals can be completed by one index.

Wrong

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

Wrong

> (4) You cannot find out their order by their FISONs.

Wrong

> (5) But you claim that only indices having FISONs are existing.

Wrong.

>
> This is a contradiction outside of matheology.

Wrong

>
>> If [r,s] is still incomplete at an
>> index that [p,q] is complete, then not all of the rationals
>> in [r,s] are finitely-indexed.
>
> You need not prove (4). It is obvious without your proof. So this is only an attempt to blur the straight argument.
>>
>> You know, it was just a few days ago that you (WM) were using
>> your claim that some intervals completed before others to
>> "prove" that dark numbers exist.[1]
>
> Of course. If (1) is assumed, then (3) is unavoidable. Then (4) proves dark indices.

Wrong, as 3 and 4 are wrong.
>>
>> Now, you're using the negation of that to prove that dark
>> numbers exist.
>
> No. Go through (1) to (5) above. Explain your position, but don't try to prove (1) to (4). It is not doubted.

Wrong. only 1 is correct,

>
> Regards, WM
>

Score 7/8 wrong = 12.5% Fail, AND you get an F--- and sanctioned for BAD MATH

On 8/19/2021 7:06 AM, WM wrote:
> Jim Burns schrieb am Mittwoch, 18. August 2021 um 23:38:55 UTC+2:
>
>
>> No finite index can complete an interval with infinitely-many
>> indexed rationals.
>
> How then can it be completed?
>
> Regards, WM
>

use infinite index, like the Natural Numbers

On 8/19/2021 8:05 AM, Gus Gassmann wrote:
> On Thursday, 19 August 2021 at 09:35:44 UTC-3, FromTheRafters wrote:
>> on 8/18/2021, Sergio supposed :
>>> On 8/18/2021 2:38 PM, WM wrote:
>>>> William schrieb am Mittwoch, 18. August 2021 um 18:32:37 UTC+2:
>>>>> On Wednesday, August 18, 2021 at 12:23:09 PM UTC-4, WM wrote:
>>>>>> William schrieb am Mittwoch, 18. August 2021 um 18:13:59 UTC+2:
>>>>>>> On Wednesday, August 18, 2021 at 7:43:08 AM UTC-4, WM wrote:
>>>>>>>> Why not? For every definable index it has meaning. If it has no meaning
>>>>>>>> later, then there are no definable indices.
>>>>>>> Nope, The fact that it has meaning for every index you can write down,
>>>>>>> does not mean it has meaning for every index you cannot write down.
>>>>>> Correct. It has no meaning for dark indices.
>>>>> The set |N_F does not contain "dark" elements.
>>>>
>>>> How do you explain that many indexes can be issued in order at the beginning
>>>> while no index can be distinguished after all intervals have been defiled?
>>>
>>> in what context ?
>>>
>>>>>
>>>>>>> The set |N_F does not contain "dark" elements.
>>>>>> But if it indexes all fraction, then it contains elements that cannot be
>>>>>> ordered.
>>>>> Nope there is no element of U_F that cannot be indexed by |N_F and no
>>>>> element of |N_F that cannot be ordered.
>>>> How do you explain that no index can be ordered after all intervals have
>>>> been defiled?
>>>
>>> in what context ?
>> Defiled? It *is* almost criminal the way he defiles those little
>> intervals.
>
> Ah, the phantasies of a lapsed physicist!
>

you can buy the clean-up type indexes for defiled intervals on Amazon.

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.

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.

It stands to reason, don't it? The German Institute for Standardization is clearly a front for evil Cantorians. (Even the addled brain of a lapsed physicist can see that...)

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.

There is a long list of common idiotic habits amongst crank down here.
One of them is to pretend to deduce significant stuff from non-specific
properties ("to be a member of a finite set" for Crank Mueckenheim, to
be a function of x and h for Crank John Gabriel). Another one is to
claim that a conventional choice is relevant (0 to be part of N or not
for Crank Mueckenheim, being called a kind number for Crank John
Gabriel).

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

On Thursday, August 19, 2021 at 8:17:25 AM UTC-4, WM wrote:
> William schrieb am Donnerstag, 19. August 2021 um 01:13:44 UTC+2:
> > On Wednesday, August 18, 2021 at 3:38:35 PM UTC-4, WM wrote:
> > > William schrieb am Mittwoch, 18. August 2021 um 18:32:37 UTC+2:
> > > > On Wednesday, August 18, 2021 at 12:23:09 PM UTC-4, WM wrote:
> > > > > William schrieb am Mittwoch, 18. August 2021 um 18:13:59 UTC+2:
> > > > > > On Wednesday, August 18, 2021 at 7:43:08 AM UTC-4, WM wrote:
> > > > >
> > > > > > > Why not? For every definable index it has meaning. If it has no meaning later, then there are no definable indices.
> > > > > > Nope, The fact that it has meaning for every index you can write down, does not mean it has meaning for every index you cannot write down.
> > > > > Correct. It has no meaning for dark indices.
> > > > The set |N_F does not contain "dark" elements.
> > > How do you explain that many indexes can be issued in order at the beginning while no index can be distinguished after all intervals have been defiled?
> > Nothing to explain as there is no problem in distinguishing elements of |N_F that cannot be written down.
> They cannot be distinguished because they cannot be written down? No that's not the true reason. If they had a discernible well-order, then this would cause contradictions.

This only causes "contradictions" if you try to use (e.g. to "complete" a bijection) a one at a time method on an infinite set.. The conclusion is that you cannot use one at a time (stepwise) methods for infinite sets. Use methods that are appropriate for infinite sets and you do not get a contradiction.

--
William Hughes

William schrieb am Donnerstag, 19. August 2021 um 18:44:34 UTC+2:
> On Thursday, August 19, 2021 at 8:17:25 AM UTC-4, WM wrote:
> > William schrieb am Donnerstag, 19. August 2021 um 01:13:44 UTC+2:
> > > On Wednesday, August 18, 2021 at 3:38:35 PM UTC-4, WM wrote:
> > > > William schrieb am Mittwoch, 18. August 2021 um 18:32:37 UTC+2:
> > > > > On Wednesday, August 18, 2021 at 12:23:09 PM UTC-4, WM wrote:
> > > > > > William schrieb am Mittwoch, 18. August 2021 um 18:13:59 UTC+2:
> > > > > > > On Wednesday, August 18, 2021 at 7:43:08 AM UTC-4, WM wrote:
> > > > > >
> > > > > > > > Why not? For every definable index it has meaning. If it has no meaning later, then there are no definable indices.
> > > > > > > Nope, The fact that it has meaning for every index you can write down, does not mean it has meaning for every index you cannot write down.
> > > > > > Correct. It has no meaning for dark indices.
> > > > > The set |N_F does not contain "dark" elements.
> > > > How do you explain that many indexes can be issued in order at the beginning while no index can be distinguished after all intervals have been defiled?
> > > Nothing to explain as there is no problem in distinguishing elements of |N_F that cannot be written down.
> > They cannot be distinguished because they cannot be written down? No that's not the true reason. If they had a discernible well-order, then this would cause contradictions.
> This only causes "contradictions" if you try to use (e.g. to "complete" a bijection) a one at a time method on an infinite set.

So it is. Step-by-step is possible and *only possible* for definable natnumbers. This is another nice definition of definable numbers.

> The conclusion is that you cannot use one at a time (stepwise) methods for infinite sets.

That is what I have been arguing for years now. Application of FISONs ceases in the darkness. It is impossible distinguish dark numbers. You can only believe that they exist, forced by the completeness of actually infinite sets.

> Use methods that are appropriate for infinite sets and you do not get a contradiction.

I only said that if |N is a sequence and its elements can be treated like terms of a sequence, then the intervals must be completed in some well-ordered way. I know that this is impossible, but most do not know this. Therefore I have devised this clear example.

Regards, WM

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.
Das System Omega bildet daher in seiner natürlichen Größenordnung eine "Folge".
Fügen wir zu dieser Folge noch als Element die 0, und zwar setzen wir sie an die erste Stelle, so erhalten wir eine Folge Omega':
0, 1, 2, 3, . . . 0, 0 + 1, . . ., ,. . .,
von welcher man sich leicht überzeugt, daß jede in ihr vorkommende Zahl Typus der Folge aller ihr vorangehenden Elemente (mit Einschluß der 0) ist. (Die Folge  hat diese Eigenschaft erst für 0 + 1). (Cantor, Brief an Dedekind, 1899)

Regards, WM

On Thursday, August 19, 2021 at 1:13:51 PM UTC-4, WM wrote:
> William schrieb am Donnerstag, 19. August 2021 um 18:44:34 UTC+2:

> > Use methods that are appropriate for infinite sets and you do not get a contradiction.
> I only said that if |N is a sequence and its elements can be treated like terms of a sequence,

Correct, but 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.

--
William Hughes

On Thursday, 19 August 2021 at 14:24:19 UTC-3, WM wrote:
> 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.
>
> Das System Omega bildet daher in seiner natürlichen Größenordnung eine "Folge".
> Fügen wir zu dieser Folge noch als Element die 0, und zwar setzen wir sie an die erste Stelle, so erhalten wir eine Folge Omega':
> 0, 1, 2, 3, . . . 0, 0 + 1, . . ., ,. . .,
> von welcher man sich leicht überzeugt, daß jede in ihr vorkommende Zahl Typus der Folge aller ihr vorangehenden Elemente (mit Einschluß der 0) ist. (Die Folge  hat diese Eigenschaft erst für 0 + 1). (Cantor, Brief an Dedekind, 1899)

So much for claiming that Cantor used the term "Reihe" instead of "Folge".

I maintain that you are too dense and too stupid to talk about infinity, have been since at least 2005, and will be to the day you die.

William schrieb am Donnerstag, 19. August 2021 um 19:24:31 UTC+2:
> On Thursday, August 19, 2021 at 1:13:51 PM UTC-4, WM wrote:
> > William schrieb am Donnerstag, 19. August 2021 um 18:44:34 UTC+2:
>
> > > Use methods that are appropriate for infinite sets and you do not get a contradiction.
> > I only said that if |N is a sequence and its elements can be treated like terms of a sequence,
> Correct, but 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. Here it is particularly obvious, but it is same for all infinite sets. That's why Cantor's bijection (which assumes that step-by-step is always possible and will never cease) fails.

If in |N the step-by-step method fails, then the bijection fails. Obviously silly results like the equinumerosity of prime numbers and algebraic numbers should teach this to every sober mind.

Regards, WM

Gus Gassmann schrieb am Donnerstag, 19. August 2021 um 19:28:13 UTC+2:
> On Thursday, 19 August 2021 at 14:24:19 UTC-3, WM wrote:

> > Obviously. It is hard to imagine any other reason for including one of the most unnatural numbers in the set of natural numbers.
> >
> > Das System Omega bildet daher in seiner natürlichen Größenordnung eine "Folge".
> > Fügen wir zu dieser Folge noch als Element die 0, und zwar setzen wir sie an die erste Stelle, so erhalten wir eine Folge Omega':
> > 0, 1, 2, 3, . . . 0, 0 + 1, . . ., ,. . .,
> > von welcher man sich leicht überzeugt, daß jede in ihr vorkommende Zahl Typus der Folge aller ihr vorangehenden Elemente (mit Einschluß der 0) ist. (Die Folge  hat diese Eigenschaft erst für 0 + 1). (Cantor, Brief an Dedekind, 1899)
> So much for claiming that Cantor used the term "Reihe" instead of "Folge"..
>
Often but not always.

Regards, WM

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

--
William Hughes