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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.

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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 11:06 UTC

Eram semper recta schrieb am Sonntag, 24. April 2022 um 09:01:20 UTC+2:
> On Saturday, 23 April 2022 at 22:26:09 UTC+3, WM wrote:

> > >
> > > 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.

Either there is no "all n" or most are dark. All you can treat is a potentially infinite collection ℕ_def

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

More is not possible. If there are all n, then we can collectively treat them

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

but that is an idea which can be wrong, Same with omega.
>
> 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).

That is clear because the limit is not reached by any definable natural argument.
>
> As for your mis-analogy f(n)=1/n, it too is problematic because for no n do we ever have 1/n = 0.

It is like it is. We cannot change it.
>
> 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?

The reason is this: The values of the function f(n) = 1/n get closer and closer to 0, although every argument n is infinitely remote from omega. But omega, if we accept it (!), is the limit of the sequence 1, 2, 3, ... . Likewise the limit of the FISONs is |N. But between every FISON and |N there are infinitely many natnumbers which cannot be treated individually:

We can reach omega by collectively treating the natural numbers

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

but not by individually treating them

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

>
> 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.

That is a misunderstanding. The limit exists but in all cases there are infinitely many dark elements between every defined element and the limit.
>
> You can retort that 0 is an actual limit whereas oo is not, however the processes are exactly the same:

Yes.
>
> To arrive at 0 wrt 1/n is independent of the number of terms,

It is impossible to arrive.

> but somehow to arrive at N depends on the number of terms.

|N is the limit, but it cannot be reached like 0 cannot be reached.

> Both reference infinity, the one with limit 0 and the other with limit ‎א‎0.

Yes.
>
> 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.

Yes.

Regards, WM

Re: Cantor's notion of countability is trash.

<aec0a94b-6586-47b5-872e-6e0c5628cb04n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=97867&group=sci.math#97867

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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 11:11 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 individually 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 individually.. I call them dark numbers.

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 11:16 UTC

WM schrieb am Sonntag, 24. April 2022 um 13:11:48 UTC+2:
> Jim Burns schrieb am Samstag, 23. April 2022 um 22:55:55 UTC+2:
Correction
This proves the existence of numbers which can only be treated collectively. I call them dark numbers.

Regards, WM

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 - Sun, 24 Apr 2022 15:28 UTC

On Sunday, April 24, 2022 at 7:14:40 AM UTC-3, WM wrote:

> We can reach omega by collectively treating the natural numbers
>
> {1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega} ,

"collectively treating" removing an infinite set.
>
> but not by individually treating

"individually treating" removing a finite set.

> 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

An infinite *set* with finite *elements*.

--
William Hughes

Re: Cantor's notion of countability is trash.

<t43rkk$1c9k$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: Sun, 24 Apr 2022 10:50:42 -0500
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 by: sergio - Sun, 24 Apr 2022 15:50 UTC

On 4/24/2022 5:14 AM, WM wrote:
> 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} ,

Errors:

1. Omega is not a natural number, so your equation is error.

>
> 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} .

Errors:
2. _def is not math,
3. Omega is not a natural number

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

Errors:
4. This is not a proof, it does not have the structure of a proof
5. Calling math mistakes "dark numbers"

>
> Regards, WM
>

5 errors in one post

Re: Cantor's notion of countability is trash.

<t43sph$a7r$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
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Subject: Re: Cantor's notion of countability is trash.
Date: Sun, 24 Apr 2022 09:10:23 -0700
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 by: FromTheRafters - Sun, 24 Apr 2022 16:10 UTC

WM laid this down on his screen :
> WM schrieb am Sonntag, 24. April 2022 um 13:11:48 UTC+2:
>> Jim Burns schrieb am Samstag, 23. April 2022 um 22:55:55 UTC+2:
> Correction
> This proves the existence of numbers which can only be treated collectively.
> I call them dark numbers.

Where do they start? Is there a 'least' dark number? Of what use are
they? Please don't confuse them with the natural numbers which can be
treated either way.

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 17:04 UTC

William schrieb am Sonntag, 24. April 2022 um 17:29:01 UTC+2:
> On Sunday, April 24, 2022 at 7:14:40 AM UTC-3, WM wrote:
>
> > We can reach omega by collectively treating the natural numbers
> >
> > {1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega} ,
> "collectively treating" removing an infinite set.
> >
> > but not by individually treating
> "individually treating" removing a finite set.

Individually concerns always elements or a finite set.

> > 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

> An infinite *set* with finite *elements*

most of which cannot be handled 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: askaske...@gmail.com (WM)
Injection-Date: Sun, 24 Apr 2022 17:05:51 +0000
Content-Type: text/plain; charset="UTF-8"
 by: WM - Sun, 24 Apr 2022 17:05 UTC

FromTheRafters schrieb am Sonntag, 24. April 2022 um 18:10:35 UTC+2:
> WM laid this down on his screen :
> > WM schrieb am Sonntag, 24. April 2022 um 13:11:48 UTC+2:
> >> Jim Burns schrieb am Samstag, 23. April 2022 um 22:55:55 UTC+2:
> > Correction
> > This proves the existence of numbers which can only be treated collectively.
> > I call them dark numbers.
> Where do they start? Is there a 'least' dark number?

No.

> Of what use are
> they? Please don't confuse them with the natural numbers which can be
> treated either way.

dark numbers can only be treated collectively.

Regards, WM

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 - Sun, 24 Apr 2022 19:19 UTC

On Sunday, April 24, 2022 at 2:04:12 PM UTC-3, WM wrote:
> William schrieb am Sonntag, 24. April 2022 um 17:29:01 UTC+2:
> > On Sunday, April 24, 2022 at 7:14:40 AM UTC-3, WM wrote:

> > > This proves the existence of
>
> > An infinite *set* with finite *elements*
> most of which cannot be handled individually.

Piffle. Each *element* is finite so each element "can be handled individually".

--
William 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 - Sun, 24 Apr 2022 19:26 UTC

On Sunday, April 24, 2022 at 2:05:55 PM UTC-3, WM wrote:
> FromTheRafters schrieb am Sonntag, 24. April 2022 um 18:10:35 UTC+2:
> > Is there a 'least' dark number?
> No.

As any non empty subset of a Peano set must contain a least element, a Peano set cannot contain a "dark" element.

--
William Hughes

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
Date: Sun, 24 Apr 2022 14:44:14 -0500
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 by: sergio - Sun, 24 Apr 2022 19:44 UTC

On 4/24/2022 2:19 PM, William wrote:
> On Sunday, April 24, 2022 at 2:04:12 PM UTC-3, WM wrote:
>> William schrieb am Sonntag, 24. April 2022 um 17:29:01 UTC+2:
>>> On Sunday, April 24, 2022 at 7:14:40 AM UTC-3, WM wrote:
>
>>>> This proves the existence of
>>
>>> An infinite *set* with finite *elements*
>> most of which cannot be handled individually.
>
> Piffle. Each *element* is finite so each element "can be handled individually".
>

each element of {1,2,3,...} is finite, "can be handled individually", and the set (of natural numbers) is infinite.

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
Date: Sun, 24 Apr 2022 22:31:25 +0200
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 by: Python - Sun, 24 Apr 2022 20:31 UTC

Crank Wolfgang Mückenheim, aka WM wrote:
> FromTheRafters schrieb am Sonntag, 24. April 2022 um 18:10:35 UTC+2:
>> WM laid this down on his screen :
>>> WM schrieb am Sonntag, 24. April 2022 um 13:11:48 UTC+2:
>>>> Jim Burns schrieb am Samstag, 23. April 2022 um 22:55:55 UTC+2:
>>> Correction
>>> This proves the existence of numbers which can only be treated collectively.
>>> I call them dark numbers.
>> Where do they start? Is there a 'least' dark number?
>
> No.
>
>> Of what use are
>> they? Please don't confuse them with the natural numbers which can be
>> treated either way.
>
> dark numbers can only be treated collectively.

Like Jews, right?

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 - Sun, 24 Apr 2022 21:55 UTC

The natural numbers are a Peano set (axiom of infinity)
A Peano set cannot contain a "dark" element
The natural numbers do not contain a "dark" element.

--
William Hughes

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Mon, 25 Apr 2022 05:30 UTC

torsdag 21 april 2022 kl. 16:16:32 UTC+2 skrev WM:
> William schrieb am Mittwoch, 20. April 2022 um 19:02:38 UTC+2:
> > On Wednesday, April 20, 2022 at 9:38:20 AM UTC-3, 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.
> > Piffle. A result you do not like is not a contradiction,
> By induction we prove that every definable natural number, i.e., every natural number which is subject to induction, belongs to a FISON but has ℵo successors which cannot be removed whatever natural number you consider. That means they cannot be used as individuals.
>
> Regards, WM

This is a pure non-sequitor

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 - Mon, 25 Apr 2022 10:18 UTC

William schrieb am Sonntag, 24. April 2022 um 21:19:40 UTC+2:
> On Sunday, April 24, 2022 at 2:04:12 PM UTC-3, WM wrote:
> > William schrieb am Sonntag, 24. April 2022 um 17:29:01 UTC+2:
> > > On Sunday, April 24, 2022 at 7:14:40 AM UTC-3, WM wrote:
>
> > > > This proves the existence of
> >
> > > An infinite *set* with finite *elements*
> > most of which cannot be handled individually.
> Piffle. Each *element* is finite so each element "can be handled individually".

Simply use logic, compare and find the difference between the isolated omega
{1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega}
and the not isolated omega
∀n ∈ ℕ_def: {1, 2, 3, ..., omega} \ {1, 2, 3, ..., n} = {n+1, n+2, ..., omega} .

All Peano natnumbers fail the first task.

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 - Mon, 25 Apr 2022 10:21 UTC

William schrieb am Sonntag, 24. April 2022 um 21:26:37 UTC+2:
> On Sunday, April 24, 2022 at 2:05:55 PM UTC-3, WM wrote:
> > FromTheRafters schrieb am Sonntag, 24. April 2022 um 18:10:35 UTC+2:
> > > Is there a 'least' dark number?
> > No.
> As any non empty subset of a Peano set must contain a least element, a Peano set cannot contain a "dark" element.

Therefore the Peano set does not contain dark elements. But between the Peano set and omega there are dark elements.

Peano natural: ∀n ∈ ℕ_P: {1, 2, 3, ..., omega} \ {1, 2, 3, ..., n} = {n+1, n+2, ..., omega} .
Dark naturals: {1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega}

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 - Mon, 25 Apr 2022 10:23 UTC

zelos...@gmail.com schrieb am Montag, 25. April 2022 um 07:30:59 UTC+2:
> torsdag 21 april 2022 kl. 16:16:32 UTC+2 skrev WM:
> > William schrieb am Mittwoch, 20. April 2022 um 19:02:38 UTC+2:
> > > On Wednesday, April 20, 2022 at 9:38:20 AM UTC-3, 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.
> > > Piffle. A result you do not like is not a contradiction,
> > By induction we prove that every definable natural number, i.e., every natural number which is subject to induction, belongs to a FISON but has ℵo successors which cannot be removed whatever natural number you consider. That means they cannot be used as individuals.

> This is a pure non-sequitor

Try logic:
Peano naturals: ∀n ∈ ℕ_def: {1, 2, 3, ..., omega} \ {1, 2, 3, ..., n} = {n+1, n+2, ..., omega} .
Dark naturals: {1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega}

Regards, WM

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 - Mon, 25 Apr 2022 15:19 UTC

On Monday, April 25, 2022 at 7:21:55 AM UTC-3, WM wrote:
> William schrieb am Sonntag, 24. April 2022 um 21:26:37 UTC+2:
> > On Sunday, April 24, 2022 at 2:05:55 PM UTC-3, WM wrote:
> > > FromTheRafters schrieb am Sonntag, 24. April 2022 um 18:10:35 UTC+2:
> > > > Is there a 'least' dark number?
> > > No.
> > As any non empty subset of a Peano set must contain a least element, a Peano set cannot contain a "dark" element.
> Therefore the Peano set does not contain dark elements.

The natural numbers are a Peano set. Thus a "dark element" cannot be a natural number.

--
William Hughes

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Tue, 26 Apr 2022 04:52 UTC

måndag 25 april 2022 kl. 12:18:24 UTC+2 skrev WM:
> William schrieb am Sonntag, 24. April 2022 um 21:19:40 UTC+2:
> > On Sunday, April 24, 2022 at 2:04:12 PM UTC-3, WM wrote:
> > > William schrieb am Sonntag, 24. April 2022 um 17:29:01 UTC+2:
> > > > On Sunday, April 24, 2022 at 7:14:40 AM UTC-3, WM wrote:
> >
> > > > > This proves the existence of
> > >
> > > > An infinite *set* with finite *elements*
> > > most of which cannot be handled individually.
> > Piffle. Each *element* is finite so each element "can be handled individually".
> Simply use logic,

It'd be bloody great if you could start using it

>compare and find the difference between the isolated omega

omega is not part of natural numbers so it is irrelevant.

> {1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega}
> and the not isolated omega
> ∀n ∈ ℕ_def: {1, 2, 3, ..., omega} \ {1, 2, 3, ..., n} = {n+1, n+2, ..., omega} .
> All Peano natnumbers fail the first task.

That again makes N_def=N

>
> Regards, WM

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Tue, 26 Apr 2022 04:53 UTC

måndag 25 april 2022 kl. 12:23:23 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Montag, 25. April 2022 um 07:30:59 UTC+2:
> > torsdag 21 april 2022 kl. 16:16:32 UTC+2 skrev WM:
> > > William schrieb am Mittwoch, 20. April 2022 um 19:02:38 UTC+2:
> > > > On Wednesday, April 20, 2022 at 9:38:20 AM UTC-3, 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.
> > > > Piffle. A result you do not like is not a contradiction,
> > > By induction we prove that every definable natural number, i.e., every natural number which is subject to induction, belongs to a FISON but has ℵo successors which cannot be removed whatever natural number you consider. That means they cannot be used as individuals.
> > This is a pure non-sequitor
> Try logic:

I am better at logic than you.

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

That applies to all of N so N_def=N, try using logic.

> Dark naturals: {1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega}

That does not define dark in any meaningful way.

>
> 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 - Tue, 26 Apr 2022 09:13 UTC

On Sunday, 24 April 2022 at 14:06:59 UTC+3, WM wrote:
> Eram semper recta schrieb am Sonntag, 24. April 2022 um 09:01:20 UTC+2:
> > On Saturday, 23 April 2022 at 22:26:09 UTC+3, WM wrote:
>
> > > >
> > > > 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.
> Either there is no "all n" or most are dark. All you can treat is a potentially infinite collection ℕ_def
> ∀n ∈ ℕ_def: {1, 2, 3, ..., omega} \ {1, 2, 3, ..., n} = {n+1, n+2, ..., omega} .
> More is not possible. If there are all n, then we can collectively treat them
> {1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega} ,
> but that is an idea which can be wrong, Same with omega.
> >
> > 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).
> That is clear because the limit is not reached by any definable natural argument.

My point exactly. The definition f'(x)=lim_{h->0} [f(x+h)-f(x)]/h is spurious.

> >
> > As for your mis-analogy f(n)=1/n, it too is problematic because for no n do we ever have 1/n = 0.
> It is like it is. We cannot change it.

"It is like it is" is not a valid argument. Nothing needs to be accepted on faith.

And of course we can define the derivative rigorously:

f'(x) = [f(x+h)-f(x)]/h - Q(x,h)

> >
> > 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?
> The reason is this: The values of the function f(n) = 1/n get closer and closer to 0, although every argument n is infinitely remote from omega. But omega, if we accept it (!), is the limit of the sequence 1, 2, 3, ... . Likewise the limit of the FISONs is |N. But between every FISON and |N there are infinitely many natnumbers which cannot be treated individually:
> We can reach omega by collectively treating the natural numbers
>
> {1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega} ,
>
> but not by individually treating them
> ∀n ∈ ℕ_def: {1, 2, 3, ..., omega} \ {1, 2, 3, ..., n} = {n+1, n+2, ..., omega} .
>
> >
> > 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.
> That is a misunderstanding. The limit exists but in all cases there are infinitely many dark elements between every defined element and the limit.
> >
> > You can retort that 0 is an actual limit whereas oo is not, however the processes are exactly the same:
> Yes.
> >
> > To arrive at 0 wrt 1/n is independent of the number of terms,
> It is impossible to arrive.
> > but somehow to arrive at N depends on the number of terms.
> |N is the limit, but it cannot be reached like 0 cannot be reached.
> > Both reference infinity, the one with limit 0 and the other with limit ‎א‎0.
> Yes.
> >
> > 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.
> Yes.
>
> 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 - Tue, 26 Apr 2022 11:19 UTC

William schrieb am Montag, 25. April 2022 um 17:19:18 UTC+2:
> On Monday, April 25, 2022 at 7:21:55 AM UTC-3, WM wrote:
> > William schrieb am Sonntag, 24. April 2022 um 21:26:37 UTC+2:
> > > On Sunday, April 24, 2022 at 2:05:55 PM UTC-3, WM wrote:
> > > > FromTheRafters schrieb am Sonntag, 24. April 2022 um 18:10:35 UTC+2:
> > > > > Is there a 'least' dark number?
> > > > No.
> > > As any non empty subset of a Peano set must contain a least element, a Peano set cannot contain a "dark" element.
> > Therefore the Peano set does not contain dark elements.
> The natural numbers are a Peano set. Thus a "dark element" cannot be a natural number.

Fact is in |N there are infinitely many natnumbers which cannot be treated individually:
We can reach omega by collectively treating the natural numbers
{1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega} ,
but not by individually treating them
∀n ∈ ℕ_def: {1, 2, 3, ..., omega} \ {1, 2, 3, ..., n} = {n+1, n+2, ..., omega} .
Simple fact.

Not all natural numbers belong to a Peano set.

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 - Tue, 26 Apr 2022 11:20 UTC

zelos...@gmail.com schrieb am Dienstag, 26. April 2022 um 06:53:26 UTC+2:
> måndag 25 april 2022 kl. 12:23:23 UTC+2 skrev WM:
> > zelos...@gmail.com schrieb am Montag, 25. April 2022 um 07:30:59 UTC+2:
> > > torsdag 21 april 2022 kl. 16:16:32 UTC+2 skrev WM:
> > > > William schrieb am Mittwoch, 20. April 2022 um 19:02:38 UTC+2:
> > > > > On Wednesday, April 20, 2022 at 9:38:20 AM UTC-3, 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.
> > > > > Piffle. A result you do not like is not a contradiction,
> > > > By induction we prove that every definable natural number, i.e., every natural number which is subject to induction, belongs to a FISON but has ℵo successors which cannot be removed whatever natural number you consider. That means they cannot be used as individuals.
> > > This is a pure non-sequitor
> > Try logic:
> I am better at logic than you.
> > Peano naturals: ∀n ∈ ℕ_def: {1, 2, 3, ..., omega} \ {1, 2, 3, ..., n} = {n+1, n+2, ..., omega} .
> That applies to all of N so N_def=N, try using logic.
> > Dark naturals: {1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega}
> That does not define dark in any meaningful way.

You are not good enough at logic.

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 - Tue, 26 Apr 2022 11:26 UTC

Eram semper recta schrieb am Dienstag, 26. April 2022 um 11:13:43 UTC+2:
> On Sunday, 24 April 2022 at 14:06:59 UTC+3, WM wrote:

> > > 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).
> > That is clear because the limit is not reached by any definable natural argument.
> My point exactly. The definition f'(x)=lim_{h->0} [f(x+h)-f(x)]/h is spurious.

It is like the definition of the limit 0 of the sequence (1/n).

Regards, WM

Re: Cantor's notion of countability is trash.

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Subject: Re: Cantor's notion of countability is trash.
From: zelos.ma...@gmail.com (zelos...@gmail.com)
Injection-Date: Tue, 26 Apr 2022 11:38:05 +0000
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 by: zelos...@gmail.com - Tue, 26 Apr 2022 11:38 UTC

tisdag 26 april 2022 kl. 11:13:43 UTC+2 skrev Eram semper recta:
> On Sunday, 24 April 2022 at 14:06:59 UTC+3, WM wrote:
> > Eram semper recta schrieb am Sonntag, 24. April 2022 um 09:01:20 UTC+2:
> > > On Saturday, 23 April 2022 at 22:26:09 UTC+3, WM wrote:
> >
> > > > >
> > > > > 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.
> > Either there is no "all n" or most are dark. All you can treat is a potentially infinite collection ℕ_def
> > ∀n ∈ ℕ_def: {1, 2, 3, ..., omega} \ {1, 2, 3, ...., n} = {n+1, n+2, ..., omega} .
> > More is not possible. If there are all n, then we can collectively treat them
> > {1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega} ,
> > but that is an idea which can be wrong, Same with omega.
> > >
> > > 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).
> > That is clear because the limit is not reached by any definable natural argument.
> My point exactly. The definition f'(x)=lim_{h->0} [f(x+h)-f(x)]/h is spurious.
> > >
> > > As for your mis-analogy f(n)=1/n, it too is problematic because for no n do we ever have 1/n = 0.
> > It is like it is. We cannot change it.
> "It is like it is" is not a valid argument. Nothing needs to be accepted on faith.
>
> And of course we can define the derivative rigorously:
>
> f'(x) = [f(x+h)-f(x)]/h - Q(x,h)
> > >
> > > 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?
> > The reason is this: The values of the function f(n) = 1/n get closer and closer to 0, although every argument n is infinitely remote from omega. But omega, if we accept it (!), is the limit of the sequence 1, 2, 3, ... .. Likewise the limit of the FISONs is |N. But between every FISON and |N there are infinitely many natnumbers which cannot be treated individually:
> > We can reach omega by collectively treating the natural numbers
> >
> > {1, 2, 3, ..., omega} \ {1, 2, 3, ...} = {omega} ,
> >
> > but not by individually treating them
> > ∀n ∈ ℕ_def: {1, 2, 3, ..., omega} \ {1, 2, 3, ...., n} = {n+1, n+2, ..., omega} .
> >
> > >
> > > 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.
> > That is a misunderstanding. The limit exists but in all cases there are infinitely many dark elements between every defined element and the limit.
> > >
> > > You can retort that 0 is an actual limit whereas oo is not, however the processes are exactly the same:
> > Yes.
> > >
> > > To arrive at 0 wrt 1/n is independent of the number of terms,
> > It is impossible to arrive.
> > > but somehow to arrive at N depends on the number of terms.
> > |N is the limit, but it cannot be reached like 0 cannot be reached.
> > > Both reference infinity, the one with limit 0 and the other with limit ‎א‎0.
> > Yes.
> > >
> > > 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.
> > Yes.
> >
> > Regards, WM
your shit is NOT rigorous

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