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tech / sci.math / Re: Cantor's notion of countability is trash.

SubjectAuthor
* Cantor's notion of countability is trash.WM
+* Re: Cantor's notion of countability is trash.sergio
|+- Re: Cantor's notion of countability is trash.sergio
|`- Re: Cantor's notion of countability is trash.Ross A. Finlayson
+- Re: Cantor's notion of countability is trash.FromTheRafters
+- Re: Cantor's notion of countability is trash.Gus Gassmann
+- Re: Cantor's notion of countability is trash.Mathin3D
+* Re: Cantor's notion of countability is trash.zelos...@gmail.com
|`* Re: Cantor's notion of countability is trash.WM
| +* Re: Cantor's notion of countability is trash.sergio
| |`* Re: Cantor's notion of countability is trash.Ross A. Finlayson
| | `* Re: Cantor's notion of countability is trash.sergio
| |  `- Re: Cantor's notion of countability is trash.Ross A. Finlayson
| +* Re: Cantor's notion of countability is trash.Gus Gassmann
| |`* Re: Cantor's notion of countability is trash.WM
| | +* Re: Cantor's notion of countability is trash.Gus Gassmann
| | |`- Re: Cantor's notion of countability is trash.Dan Christensen
| | `- Re: Cantor's notion of countability is trash.sergio
| `- Re: Cantor's notion of countability is trash.zelos...@gmail.com
+- Re: Cantor's notion of countability is trash.Eram semper recta
+* Re: Cantor's notion of countability is trash.Dan Christensen
|`* Re: Cantor's notion of countability is trash.WM
| +- Re: Cantor's notion of countability is trash.sergio
| +- Re: Cantor's notion of countability is trash.Dan Christensen
| +* Re: Cantor's notion of countability is trash.Eram semper recta
| |+- Re: Cantor's notion of countability is trash.zelos...@gmail.com
| |+- STUDENTS BEWARE: Don't be a victim of JG's fake mathDan Christensen
| |`* Re: Cantor's notion of countability is trash.WM
| | +- Re: Cantor's notion of countability is trash.FromTheRafters
| | +- Re: Cantor's notion of countability is trash.sergio
| | +- Re: Cantor's notion of countability is trash.Eram semper recta
| | `* Re: Cantor's notion of countability is trash.zelos...@gmail.com
| |  `* Re: Cantor's notion of countability is trash.WM
| |   +- Re: Cantor's notion of countability is trash.FromTheRafters
| |   `* Re: Cantor's notion of countability is trash.sergio
| |    +* Re: Cantor's notion of countability is trash.Ross A. Finlayson
| |    |`* Re: Cantor's notion of countability is trash.sergio
| |    | `- Re: Cantor's notion of countability is trash.Ross A. Finlayson
| |    `- Re: Cantor's notion of countability is trash.Takabe Matsumura
| `- Re: Cantor's notion of countability is trash.zelos...@gmail.com
+* Re: Cantor's notion of countability is trash.Eram semper recta
|+- Re: Cantor's notion of countability is trash.zelos...@gmail.com
|+* Re: Cantor's notion of countability is trash.WM
||+* Re: Cantor's notion of countability is trash.FromTheRafters
|||`* Re: Cantor's notion of countability is trash.WM
||| `* Re: Cantor's notion of countability is trash.FromTheRafters
|||  `* Re: Cantor's notion of countability is trash.WM
|||   +* Re: Cantor's notion of countability is trash.sergio
|||   |+- Re: Cantor's notion of countability is trash.Ross A. Finlayson
|||   |`* Re: Cantor's notion of countability is trash.WM
|||   | +* Re: Cantor's notion of countability is trash.Gus Gassmann
|||   | |`* Re: Cantor's notion of countability is trash.WM
|||   | | +* Re: Cantor's notion of countability is trash.Gus Gassmann
|||   | | |`* Re: Cantor's notion of countability is trash.WM
|||   | | | +* Re: Cantor's notion of countability is trash.Eram semper recta
|||   | | | |`- Re: Cantor's notion of countability is trash.Eram semper recta
|||   | | | +- Re: Cantor's notion of countability is trash.Gus Gassmann
|||   | | | +- Re: Cantor's notion of countability is trash.Russ Diaz
|||   | | | `- Re: Cantor's notion of countability is trash.sergio
|||   | | `* Re: Cantor's notion of countability is trash.sergio
|||   | |  `* Re: Cantor's notion of countability is trash.WM
|||   | |   `- Re: Cantor's notion of countability is trash.sergio
|||   | +* Re: Cantor's notion of countability is trash.sergio
|||   | |`* Re: Cantor's notion of countability is trash.FromTheRafters
|||   | | +* Re: Cantor's notion of countability is trash.Ross A. Finlayson
|||   | | |`* Re: Cantor's notion of countability is trash.FromTheRafters
|||   | | | +* Re: Cantor's notion of countability is trash.sergio
|||   | | | |`* Re: Cantor's notion of countability is trash.FredJeffries
|||   | | | | `- Re: Cantor's notion of countability is trash.sergio
|||   | | | `* Re: Cantor's notion of countability is trash.Ross A. Finlayson
|||   | | |  `- Re: Cantor's notion of countability is trash.sergio
|||   | | `- Re: Cantor's notion of countability is trash.sergio
|||   | `* Re: Cantor's notion of countability is trash.zelos...@gmail.com
|||   |  `* Re: Cantor's notion of countability is trash.WM
|||   |   +* Re: Cantor's notion of countability is trash.FromTheRafters
|||   |   |`* Re: Cantor's notion of countability is trash.WM
|||   |   | `* Re: Cantor's notion of countability is trash.FromTheRafters
|||   |   |  `* Re: Cantor's notion of countability is trash.WM
|||   |   |   +- Re: Cantor's notion of countability is trash.sergio
|||   |   |   `- Re: Cantor's notion of countability is trash.Eram semper recta
|||   |   +- Re: Cantor's notion of countability is trash.sergio
|||   |   `* Re: Cantor's notion of countability is trash.Alan Mackenzie
|||   |    `* Re: Cantor's notion of countability is trash.WM
|||   |     `- Re: Cantor's notion of countability is trash.sergio
|||   +* Re: Cantor's notion of countability is trash.Eram semper recta
|||   |+* Re: Cantor's notion of countability is trash.Eram semper recta
|||   ||+* Re: Cantor's notion of countability is trash.Eram semper recta
|||   |||`* Re: Cantor's notion of countability is trash.Eram semper recta
|||   ||| `- Re: Cantor's notion of countability is trash.Eram semper recta
|||   ||`* Re: Cantor's notion of countability is trash.zelos...@gmail.com
|||   || `* Re: Cantor's notion of countability is trash.Takabe Matsumura
|||   ||  `- Re: Cantor's notion of countability is trash.zelos...@gmail.com
|||   |`- Re: Cantor's notion of countability is trash.zelos...@gmail.com
|||   +* Re: Cantor's notion of countability is trash.Jim Burns
|||   |+* Re: Cantor's notion of countability is trash.Ross A. Finlayson
|||   ||`- Re: Cantor's notion of countability is trash.Takabe Matsumura
|||   |`* Re: Cantor's notion of countability is trash.WM
|||   | +- Re: Cantor's notion of countability is trash.sergio
|||   | +- Re: Cantor's notion of countability is trash.Takabe Matsumura
|||   | +* Re: Cantor's notion of countability is trash.Jim Burns
|||   | |`* Re: Cantor's notion of countability is trash.WM
|||   | `* Re: Cantor's notion of countability is trash.Eram semper recta
|||   `- Re: Cantor's notion of countability is trash.zelos...@gmail.com
||`- Re: Cantor's notion of countability is trash.zelos...@gmail.com
|`- Re: Cantor's notion of countability is trash.Mathin3D
+* Bullshit baffles brains.Eram semper recta
+* Re: Cantor's notion of countability is trash.David Petry
`* Re: Cantor's notion of countability is trash.mitchr...@gmail.com

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Re: Cantor's notion of countability is trash.

<t3utt9$msp$1@dont-email.me>

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https://www.novabbs.com/tech/article-flat.php?id=97748&group=sci.math#97748

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Cantor's notion of countability is trash.
Date: Fri, 22 Apr 2022 11:58:43 -0700
Organization: Peripheral Visions
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 by: FromTheRafters - Fri, 22 Apr 2022 18:58 UTC

WM was thinking very hard :
> zelos...@gmail.com schrieb am Dienstag, 19. April 2022 um 09:38:32 UTC+2:
>> torsdag 14 april 2022 kl. 13:30:41 UTC+2 skrev WM:
>>> William schrieb am Mittwoch, 13. April 2022 um 23:41:11 UTC+2:
>>>> On Wednesday, April 13, 2022 at 5:22:05 PM UTC-3, WM wrote:
>>>>> William schrieb am Mittwoch, 13. April 2022 um 21:07:30 UTC+2:
>>>>> You can only notice the absence of natnumbers in the intersection
>>>> And thus, be definition of the intersection, the intersection in empty.
>>> But that is contrary to mathematics.
>>
>> Nope, it is according to mathematics.
>
> Non-empty endsegments

Since these endsegments are inductive sets, they are all non-empty.

Re: Cantor's notion of countability is trash.

<t3uu0l$1aah$1@gioia.aioe.org>

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From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Cantor's notion of countability is trash.
Date: Fri, 22 Apr 2022 14:00:35 -0500
Organization: Aioe.org NNTP Server
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 by: sergio - Fri, 22 Apr 2022 19:00 UTC

On 4/22/2022 1:36 PM, WM wrote:
> horand....@gmail.com schrieb am Freitag, 22. April 2022 um 18:58:19 UTC+2:
>> On Friday, 22 April 2022 at 11:41:55 UTC-3, WM wrote:
>> [...]
>>> A function or sequence f(n) has domain n ∈ ℕ. It cannot be discontinuous at ω because there is no value to be attached at ω. ω does not belong to ℕ. f(ω) can only be the limit calculated from the function for n --> oo.
>> If f(.) is the card(.) function, then your method will go wrong.
>
> There is no card function.

yes there is a card function.

> If you go through all definable n you will never reach ω

"definable" is your fake math.

>
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .

"_def" is your fake math.

>
> If you remove all n collectively, only ω remains.
>
> |ℕ \ {1, 2, 3, ...}| = 0 .

you state "only ω remains", so where is it ? did it go on vacation ?

>
> Hence most natnumbers are dark

Wrong, you call them "dark" because you do not have the Math skills.

> Regards, WM

Re: Cantor's notion of countability is trash.

<t3uu30$1aah$2@gioia.aioe.org>

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From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Cantor's notion of countability is trash.
Date: Fri, 22 Apr 2022 14:01:51 -0500
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 by: sergio - Fri, 22 Apr 2022 19:01 UTC

On 4/22/2022 1:40 PM, WM wrote:
> Eram semper recta schrieb am Freitag, 22. April 2022 um 09:01:00 UTC+2:
>
>> Your argument wrt the topic is correct. However, you cannot see that your alternate belief wrt the mainstream derivative is false and can be rejected in the same way
>
> I do not reject the existence of a limit like 0 of the sequence (1/n). What I reject is the definability of all terms 1/n.
>
> Regards, WM

your "definability" is quack math, and is rejected as "bad math".

Re: Cantor's notion of countability is trash.

<t3uucu$1g1o$1@gioia.aioe.org>

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From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Cantor's notion of countability is trash.
Date: Fri, 22 Apr 2022 14:07:09 -0500
Organization: Aioe.org NNTP Server
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 by: sergio - Fri, 22 Apr 2022 19:07 UTC

On 4/22/2022 1:45 PM, WM wrote:
> William schrieb am Freitag, 22. April 2022 um 19:05:11 UTC+2:
>> There are no elements of N_p which follow the *set* of FISONs. No Peano set contains a "dark" element
>
> We cannot remove all natnumbers by removing them individually, for instance their FISONs, because all are not sufficient:
>
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>
> But we can remove all natnumbers collectively.
>
> |ℕ \ {1, 2, 3, ...}| = 0 .
>
> That proves a difference between individually and collectively treating numbers.
> Regards, WM

your conclusion is wrong, because your thinking is wrong.

one equation stops at n, the other does not.

"removing", "individually", "collectively", "treating" are not math terms. You use them because you do not know the language of math, you do not think
in Math.

Re: Cantor's notion of countability is trash.

<acd2b798-4893-4be2-b5a6-fbf5c07db6f4n@googlegroups.com>

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Subject: Re: Cantor's notion of countability is trash.
From: fredjeff...@gmail.com (FredJeffries)
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 by: FredJeffries - Fri, 22 Apr 2022 19:23 UTC

On Friday, April 22, 2022 at 9:58:19 AM UTC-7, horand....@gmail.com wrote:
> On Friday, 22 April 2022 at 11:41:55 UTC-3, WM wrote:
> [...]
> > A function or sequence f(n) has domain n ∈ ℕ. It cannot be discontinuous at ω because there is no value to be attached at ω. ω does not belong to ℕ.

Thus we have: A function CANNOT be discontinuous at a point where it is not defined. Therefore, every function MUST be CONTINUOUS at every point at which it is not defined!

marvelous.

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: horand.g...@gmail.com (Gus Gassmann)
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 by: Gus Gassmann - Fri, 22 Apr 2022 19:42 UTC

On Friday, 22 April 2022 at 15:36:27 UTC-3, WM wrote:
> horand....@gmail.com schrieb am Freitag, 22. April 2022 um 18:58:19 UTC+2:
> > On Friday, 22 April 2022 at 11:41:55 UTC-3, WM wrote:
> > [...]
> > > A function or sequence f(n) has domain n ∈ ℕ. It cannot be discontinuous at ω because there is no value to be attached at ω. ω does not belong to ℕ. f(ω) can only be the limit calculated from the function for n --> oo.
> > If f(.) is the card(.) function, then your method will go wrong.
> There is no card function.

Piss off. You throw around a function f that you never define. I ask about it, you get snippy. AND you have no clue what continuity means.

> If you go through all definable n you will never reach ω
>
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
>
> If you remove all n collectively, only ω remains.

Are you NUTS? *NOTHING* remains, you blithering idiot. ω is not a natural number.

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
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 by: sergio - Fri, 22 Apr 2022 19:42 UTC

On 4/19/2022 10:41 AM, WM wrote:
> zelos...@gmail.com schrieb am Dienstag, 19. April 2022 um 09:38:32 UTC+2:
>> torsdag 14 april 2022 kl. 13:30:41 UTC+2 skrev WM:
>>> William schrieb am Mittwoch, 13. April 2022 um 23:41:11 UTC+2:
>>>> On Wednesday, April 13, 2022 at 5:22:05 PM UTC-3, WM wrote:
>>>>> William schrieb am Mittwoch, 13. April 2022 um 21:07:30 UTC+2:
>>>>
>>>>> You can only notice the absence of natnumbers in the intersection
>>>> And thus, be definition of the intersection, the intersection in empty.
>>> But that is contrary to mathematics.
>>
>> Nope, it is according to mathematics.
>
> Non-empty endsegments have a nonempty intersection with all their precedessors. All definable endsegments are predecessors of non-empty endsegments. This fact cannot be countered by quantifier magic.
>
> Regards, WM

trivial.

all endsegments are infinite sets.

there are no empty endsegments.

the intersection of all endsegments is not an endsegment and is empty.

next....

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: askaske...@gmail.com (WM)
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 by: WM - Fri, 22 Apr 2022 19:47 UTC

FromTheRafters schrieb am Freitag, 22. April 2022 um 20:59:01 UTC+2:

> Since these endsegments are inductive sets, they are all non-empty.

Then they are less than ℵo.
It is impossible to divide |N into two consecutive sets the first of which is infinite.
It is possible that there are finitely many endsegments with ℵo-infinite contents larger than all indices of endsegments.
It is possible that there are ℵo-infinitely many endsegments having only contents which is also used as indices of the endsegements.
It is impossible to have ℵo-infinitely many ℵo-infinite endsegements.

Regards, WM

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: askaske...@gmail.com (WM)
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 by: WM - Fri, 22 Apr 2022 19:50 UTC

FredJeffries schrieb am Freitag, 22. April 2022 um 21:24:04 UTC+2:
> On Friday, April 22, 2022 at 9:58:19 AM UTC-7, horand....@gmail.com wrote:
> > On Friday, 22 April 2022 at 11:41:55 UTC-3, WM wrote:
> > [...]
> > > A function or sequence f(n) has domain n ∈ ℕ. It cannot be discontinuous at ω because there is no value to be attached at ω. ω does not belong to ℕ.
> Thus we have: A function CANNOT be discontinuous at a point where it is not defined. Therefore, every function MUST be CONTINUOUS at every point at which it is not defined!

The limit (usually) is not part of the function.

Regards, WM

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: askaske...@gmail.com (WM)
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 by: WM - Fri, 22 Apr 2022 19:53 UTC

horand....@gmail.com schrieb am Freitag, 22. April 2022 um 21:42:20 UTC+2:
> On Friday, 22 April 2022 at 15:36:27 UTC-3, WM wrote:

> > If you go through all definable n you will never reach ω
> >
> > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .

If you remove all n collectively from {ℕ, ω} only ω remains.

Regards, WM

Re: Cantor's notion of countability is trash.

<t3v1bo$ovm$1@gioia.aioe.org>

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From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Cantor's notion of countability is trash.
Date: Fri, 22 Apr 2022 14:57:42 -0500
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 by: sergio - Fri, 22 Apr 2022 19:57 UTC

On 4/22/2022 2:47 PM, WM wrote:
> FromTheRafters schrieb am Freitag, 22. April 2022 um 20:59:01 UTC+2:
>
>> Since these endsegments are inductive sets, they are all non-empty.
>
> Then they are less than ℵo.

not so, all endsegments are infinite sets. If you do not think so, show your proof.

> It is impossible to divide |N into two consecutive sets the first of which is infinite.

if you divide it into 3 consecutive sets, the last is infinite.

> It is possible that there are finitely many endsegments with ℵo-infinite contents larger than all indices of endsegments.

So, what is larger than the indices (elements) in N ?

any answers ??

> It is possible that there are ℵo-infinitely many endsegments having only contents which is also used as indices of the endsegements.

too general, be specific.

> It is impossible to have ℵo-infinitely many ℵo-infinite endsegements.

Wrong.

>
> Regards, WM

4 mistakes in one post.

Re: Cantor's notion of countability is trash.

<t3v81f$17u8$1@gioia.aioe.org>

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From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Cantor's notion of countability is trash.
Date: Fri, 22 Apr 2022 16:51:42 -0500
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 by: sergio - Fri, 22 Apr 2022 21:51 UTC

On 4/22/2022 2:53 PM, WM wrote:
> horand....@gmail.com schrieb am Freitag, 22. April 2022 um 21:42:20 UTC+2:
>> On Friday, 22 April 2022 at 15:36:27 UTC-3, WM wrote:
>
>>> If you go through all definable n you will never reach ω
>>>
>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
>
> If you remove all n collectively from {ℕ, ω} only ω remains.
>
> Regards, WM

nope, that is not right either, try again, and pay attention to Math Nomenclature.

Re: Cantor's notion of countability is trash.

<t3v8dp$1fdc$1@gioia.aioe.org>

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From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Cantor's notion of countability is trash.
Date: Fri, 22 Apr 2022 16:58:15 -0500
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 by: sergio - Fri, 22 Apr 2022 21:58 UTC

On 4/22/2022 2:50 PM, WM wrote:
> FredJeffries schrieb am Freitag, 22. April 2022 um 21:24:04 UTC+2:
>> On Friday, April 22, 2022 at 9:58:19 AM UTC-7, horand....@gmail.com wrote:
>>> On Friday, 22 April 2022 at 11:41:55 UTC-3, WM wrote:
>>> [...]
>>>> A function or sequence f(n) has domain n ∈ ℕ. It cannot be discontinuous at ω because there is no value to be attached at ω. ω does not belong to ℕ.
>> Thus we have: A function CANNOT be discontinuous at a point where it is not defined. Therefore, every function MUST be CONTINUOUS at every point at which it is not defined!
>
> The limit (usually) is not part of the function.
>
> Regards, WM

I suggest you forget using generalizations.

Extra Credit- show what is wrong with each of the following statements;

1. "A function or sequence"

2. "f(n) has domain n ∈ ℕ"

3. "It cannot be discontinuous at ω because there is no value to be attached at ω."

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: horand.g...@gmail.com (Gus Gassmann)
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 by: Gus Gassmann - Fri, 22 Apr 2022 22:49 UTC

On Friday, 22 April 2022 at 16:53:15 UTC-3, WM wrote:
[...]
> If you remove all n collectively from {ℕ, ω} only ω remains.

Congratulations on this earth-shattering insight. Only, where the fuck did you get the notion that anyone was talking about {ℕ, ω}?

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: wpihug...@gmail.com (William)
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 by: William - Sat, 23 Apr 2022 00:23 UTC

On Friday, April 22, 2022 at 3:45:53 PM UTC-3, WM wrote:
> William schrieb am Freitag, 22. April 2022 um 19:05:11 UTC+2:
> > There are no elements of N_p which follow the *set* of FISONs. No Peano set contains a "dark" element
> We cannot remove all natnumbers by removing them individually,

Whoosh, The sound of WM changing the subject. By "removing them individually" you mean remove an infinite set. You are correct. One cannot remove all elements of N_p without removing an infinite set.

--
Willliam Hughes

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: wpihug...@gmail.com (William)
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 by: William - Sat, 23 Apr 2022 00:38 UTC

On Friday, April 22, 2022 at 3:45:53 PM UTC-3, WM wrote:
> William schrieb am Freitag, 22. April 2022 um 19:05:11 UTC+2:
> > There are no elements of N_p which follow the *set* of FISONs. No Peano set contains a "dark" element
> We cannot remove all natnumbers by removing them individually,

Whoosh, The sound of WM changing the subject. By "removing them individually" you mean remove a finite set. You are correct. One cannot remove the elements of N_p without removing an infinite set. How you remove N_p has nothing to do with the questions of whether the *set* N_p is followed by an element of N_p, or whether a Peano set contains a "dark" element.

--
William Hughes

Re: Cantor's notion of countability is trash.

<132d9eb0-5358-4bbd-bfeb-8db74ceb0100n@googlegroups.com>

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Subject: Re: Cantor's notion of countability is trash.
From: askaske...@gmail.com (WM)
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 by: WM - Sat, 23 Apr 2022 12:52 UTC

William schrieb am Samstag, 23. April 2022 um 02:38:52 UTC+2:
> On Friday, April 22, 2022 at 3:45:53 PM UTC-3, WM wrote:
> > William schrieb am Freitag, 22. April 2022 um 19:05:11 UTC+2:
> > > There are no elements of N_p which follow the *set* of FISONs. No Peano set contains a "dark" element
> > We cannot remove all natnumbers by removing them individually,
> By "removing them individually" you mean remove a finite set.

Every finite set, but with no upper threshold, that means more than every given finite number. That is potentially infinite. Other removals are only possible collectively.

> You are correct. One cannot remove the elements of N_p without removing an infinite set. How you remove N_p has nothing to do with the questions of whether the *set* N_p is followed by an element of N_p, or whether a Peano set contains a "dark" element.

By definition: dark is all that you cannot remove individually.

Regards, WM

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: thenewca...@gmail.com (Eram semper recta)
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 by: Eram semper recta - Sat, 23 Apr 2022 13:27 UTC

On Saturday, 23 April 2022 at 15:52:51 UTC+3, WM wrote:
> William schrieb am Samstag, 23. April 2022 um 02:38:52 UTC+2:
> > On Friday, April 22, 2022 at 3:45:53 PM UTC-3, WM wrote:
> > > William schrieb am Freitag, 22. April 2022 um 19:05:11 UTC+2:
> > > > There are no elements of N_p which follow the *set* of FISONs. No Peano set contains a "dark" element
> > > We cannot remove all natnumbers by removing them individually,
> > By "removing them individually" you mean remove a finite set.
> Every finite set, but with no upper threshold, that means more than every given finite number. That is potentially infinite. Other removals are only possible collectively.
> > You are correct. One cannot remove the elements of N_p without removing an infinite set. How you remove N_p has nothing to do with the questions of whether the *set* N_p is followed by an element of N_p, or whether a Peano set contains a "dark" element.
> By definition: dark is all that you cannot remove individually.

Even simpler: One cannot remove that which one does not have.

>
> Regards, WM

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: thenewca...@gmail.com (Eram semper recta)
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 by: Eram semper recta - Sat, 23 Apr 2022 13:34 UTC

On Friday, 22 April 2022 at 21:41:02 UTC+3, WM wrote:
> Eram semper recta schrieb am Freitag, 22. April 2022 um 09:01:00 UTC+2:
>
> > Your argument wrt the topic is correct. However, you cannot see that your alternate belief wrt the mainstream derivative is false and can be rejected in the same way
> I do not reject the existence of a limit like 0 of the sequence (1/n). What I reject is the definability of all terms 1/n.

Not what I said. I don't know why you removed my comment? That is not the right thing to do.

Here's what I wrote:

One can say the same about the flawed mainstream definition of derivative:

f'(x) = lim_{h->0} [f(x+h)-f(x)]/h

As h decreases indefinitely, the finite differences approach a "limit":

[f(x+h_1)-f(x)]/h_1 ; [f(x+h_2)-f(x)]/h_2; ... ; [f(x+h_n)-f(x)]/h_n; ...

The difference between [f(x+h_n)-f(x)]/h_n and the "ultimate ratio" [f(x+h_ω )-f(x)]/h_ω
becomes smaller than every desired ε: |[f(x+h_n)-f(x)]/h_n - [f(x+h_ω )-f(x)]/h_ω|<ε.
But the number of terms between n and ω remains "infinite" |ω-n|=א‎0.

While you are busy with your epsilonic arguments, you fail to see that there is
no ratio [f(x+h_ω )-f(x)]/h_ω that can be the limit (derivative) and yet the derivative
itself is a *ratio of magnitudes*, viz. rise/run of tangent line.

So it's not a case of you rejecting the limit, but rather the opposite - you accept the
limit in spite of the fact that the number of terms between n and ω remains "infinite" |ω-n|=א‎0.

This is your hypocrisy.

Most of the above terms are unknown (dark?).

>
> Regards, WM

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: askaske...@gmail.com (WM)
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 by: WM - Sat, 23 Apr 2022 19:26 UTC

Eram semper recta schrieb am Samstag, 23. April 2022 um 15:34:38 UTC+2:

> The difference between [f(x+h_n)-f(x)]/h_n and the "ultimate ratio" [f(x+h_ω )-f(x)]/h_ω
> becomes smaller than every desired ε: |[f(x+h_n)-f(x)]/h_n - [f(x+h_ω )-f(x)]/h_ω|<ε.
> But the number of terms between n and ω remains "infinite" |ω-n|=א‎0.

Yes.
>
> While you are busy with your epsilonic arguments, you fail to see that there is
> no ratio [f(x+h_ω )-f(x)]/h_ω that can be the limit (derivative) and yet the derivative
> itself is a *ratio of magnitudes*, viz. rise/run of tangent line.
>
> So it's not a case of you rejecting the limit, but rather the opposite - you accept the
> limit in spite of the fact that the number of terms between n and ω remains "infinite" |ω-n|=א‎0.

Yes. Every n that we can define satisfies the criterion.
>
> This is your hypocrisy.
>
> Most of the above terms are unknown (dark?).

Most are dark. But by the general rule f(n) = 1/n we know that we can get as closely as we like to zero while the dark terms will never revert the result.

Regards, WM

Re: Cantor's notion of countability is trash.

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From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Cantor's notion of countability is trash.
Date: Sat, 23 Apr 2022 14:45:53 -0500
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 by: sergio - Sat, 23 Apr 2022 19:45 UTC

On 4/23/2022 7:52 AM, WM wrote:
> William schrieb am Samstag, 23. April 2022 um 02:38:52 UTC+2:
>> On Friday, April 22, 2022 at 3:45:53 PM UTC-3, WM wrote:
>>> William schrieb am Freitag, 22. April 2022 um 19:05:11 UTC+2:
>>>> There are no elements of N_p which follow the *set* of FISONs. No Peano set contains a "dark" element
>>> We cannot remove all natnumbers by removing them individually,
>> By "removing them individually" you mean remove a finite set.
>
> Every finite set, but with no upper threshold, that means more than every given finite number.

what number is more than every finite number ?

> That is potentially infinite.

no, that is called finite.

> Other removals are only possible collectively.

Collective Removal is what happens to garbage.

>
>> You are correct. One cannot remove the elements of N_p without removing an infinite set. How you remove N_p has nothing to do with the questions of whether the *set* N_p is followed by an element of N_p, or whether a Peano set contains a "dark" element.
>
> By definition: dark is all that you cannot remove individually.

wrong, there are no dark numbers, nor dark sets, nor dark elements, that darkness is all in your mind.

You need to mathematically Prove dark numbers exist. You have failed to do so.

if dark numbers existed, how would you know ?

>
> Regards, WM

Re: Cantor's notion of countability is trash.

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Cantor's notion of countability is trash.
Date: Sat, 23 Apr 2022 16:55:45 -0400
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 by: Jim Burns - Sat, 23 Apr 2022 20:55 UTC

On 4/20/2022 8:38 AM, WM wrote:
> William schrieb
> am Dienstag, 19. April 2022 um 18:04:38 UTC+2:

>> A "potentially infinite set" is nonsense.
>
> Alas it is the only set that we can apply in mathematics.
>
>> We apply induction exactly once,
>> and show something about each of
>> the infinitely many elements of N_p.
>
> The most important result of induction is this:
> Every number covered by induction belongs to
> a finite set which is followed by an infinite set of
> dark numbers, ℵ₀ of which will remain dark.

The most important result of induction is this:

Let P(i) be a predicate of second ends of
collections with counting-orders which
begin at 1 and end somewhere.

If
any second end k exists such that P(1) and ~P(k)
then
the second ends from 1 to k can be split into
BEFORE and AFTER such that
if ~P(i) or i is after i' where ~P(i')
then i is in AFTER
otherwise, i is in BEFORE.

There must be a step, somewhere, from BEFORE to AFTER.
There must be j in BEFORE and j+1 in AFTER
.... because _counting-order_

Also, because of the way BEFORE and AFTER are defined,
P(j) and ~P(j+1)

Therefore,
if
second end k exists such that P(1) and ~P(k)
then
second end j exists such that P(j) and ~P(j+1)

That's induction.

In its more familiar but equivalent guise,
if
P(1) and,
for each second end j, if P(j) then P(j+1),
then
for each second end k, P(k)

There are infinitely-many second ends in all.
But induction works because there are _finitely-many_
second ends between P(1) and ~P(k),
forcing some step to be from P(j) to ~P(j+1)
No such step implies no ~P(k)

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: thenewca...@gmail.com (Eram semper recta)
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 by: Eram semper recta - Sun, 24 Apr 2022 07:01 UTC

On Saturday, 23 April 2022 at 22:26:09 UTC+3, WM wrote:
> Eram semper recta schrieb am Samstag, 23. April 2022 um 15:34:38 UTC+2:
>
> > The difference between [f(x+h_n)-f(x)]/h_n and the "ultimate ratio" [f(x+h_ω )-f(x)]/h_ω
> > becomes smaller than every desired ε: |[f(x+h_n)-f(x)]/h_n - [f(x+h_ω )-f(x)]/h_ω|<ε.
> > But the number of terms between n and ω remains "infinite" |ω-n|=א‎0.
> Yes.
> >
> > While you are busy with your epsilonic arguments, you fail to see that there is
> > no ratio [f(x+h_ω )-f(x)]/h_ω that can be the limit (derivative) and yet the derivative
> > itself is a *ratio of magnitudes*, viz. rise/run of tangent line.
> >
> > So it's not a case of you rejecting the limit, but rather the opposite - you accept the
> > limit in spite of the fact that the number of terms between n and ω remains "infinite" |ω-n|=א‎0.
> Yes. Every n that we can define satisfies the criterion.

But not all n. It's similar to your FISON argument. Every FISON is finite but not ALL. Therefore, this kind of reasoning suffers from the same problems.

> >
> > This is your hypocrisy.
> >
> > Most of the above terms are unknown (dark?).
> Most are dark. But by the general rule f(n) = 1/n we know that we can get as closely as we like to zero while the dark terms will never revert the result.

....which isn't very reassuring because we don't care about the general rule but whether such a ratio (finite difference) is possible for ALL n.

One is forced to accept that such an "ultimate ratio" exists based on faith.. In actual fact, none of the n results in [f(x+h_n)-f(x)]/h_n ever being f'(x).

As for your mis-analogy f(n)=1/n, it too is problematic because for no n do we ever have 1/n = 0. This is the result of mainstream academics accepting the dysfunctional thinking of Euler.

One can counter your argument using this very example.

How is it you accept that f(n)=1/n has a limit of 0 as n increases indefinitely and yet think that the limit as n increases indefinitely in a FISON will not result in the set of natural numbers?

It's clear that you accept 1/n will have a limit of 0 as n->oo, but for some reason [1,2,3, ... n] will not have a limit as n->oo.

You can retort that 0 is an actual limit whereas oo is not, however the processes are exactly the same:

To arrive at 0 wrt 1/n is independent of the number of terms, but somehow to arrive at N depends on the number of terms. Clearly we have a biased approach here. Both reference infinity, the one with limit 0 and the other with limit ‎א‎0.

Just as there is no n that satisfies 1/n = 0, there is no h which satisfies [f(x+h)-f(x)]/h = f'(x) unless f is a straight line.

>
> Regards, WM

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: askaske...@gmail.com (WM)
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 by: WM - Sun, 24 Apr 2022 10:14 UTC

sergio schrieb am Samstag, 23. April 2022 um 21:46:03 UTC+2:

> if dark numbers existed, how would you know ?

We can reach omega by collectively treating the natural numbers

{1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega} ,

but not by individually treating them. For all definable n we get

∀n ∈ ℕ_def: {1, 2, 3, ..., omega} \ {1, 2, 3, ..., n} = {n+1, n+2, ..., omega} .

This proves the existence of numbers which can only be treated collectively.. I call them dark numbers.

Regards, WM

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Date: Sun, 24 Apr 2022 03:17:20 -0700 (PDT)
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Subject: Re: Cantor's notion of countability is trash.
From: askaske...@gmail.com (WM)
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 by: WM - Sun, 24 Apr 2022 10:17 UTC

Jim Burns schrieb am Samstag, 23. April 2022 um 22:55:55 UTC+2:

> There are infinitely-many second ends in all.

Potentially infinitely many.

But we cannot reach omega and ℵ₀ by collectively treating the natural numbers

∀n ∈ ℕ_def: {1, 2, 3, ..., omega} \ {1, 2, 3, ..., n} = {n+1, n+2, ..., omega} .

This proves the existence of numbers which can only be treated collectively.. I call them dark numbers.

Regards, WM

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