söndag 17 oktober 2021 kl. 15:00:10 UTC+2 skrev WM:
> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 14:53:42 UTC+2:
> > WM formulated on Sunday :
> > > FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 20:50:14 UTC+2:
> > >> WM laid this down on his screen :
> > >
> > >>>>> there are not more than four endsegments containing the natural
> > >>>>> numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
> > >
> > >> I understand *and* disagree. I am not alone in this.
> > >
> > > Really? Who else beliefs that there are more than four endsegments?
> > Nice editing job. You know of course that I was disagreeing with your
> > "definable" and "dark" weasel-word numbers in the rest of your post.
> No, how should I? If you understand that above only four can exist, then you could also understand that only n can exist containing {n, n+1, n+2, ....}. Then you could also understand that this is true for every n starting an infinite set. What is the difference?
>
> Regards, WM

Yes, it is weasel words you use to avoid having to deal with the precision of mathematics because you know you are wrong.

WM formulated on Sunday :
> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 14:44:19 UTC+2:
>> WM wrote :
>
>>> (C) There are not more than finitely many infinite endsegments.
>> A declaration with nothing to back it up. Each natural number has such
>> a following, no exceptions.
>
> Each definable natural number

Which one isn't?

WM presented the following explanation :
> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 14:44:19 UTC+2:

> Each infinite endsegment

Which one isn't?

On Saturday, 16 October 2021 at 15:50:14 UTC-3, FromTheRafters wrote:
[...]
> > Fact is: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo and |ℕ \ {1, 2, 3, ...}| =
> > 0. Whether you agree and understand or disagree and not understand is
> > irrelevant.
> I understand *and* disagree. I am not alone in this.

Wait. I agree with these statements. What I (and I hope, you) disagree with is that this has any significance beyond the rather trivial fact that both statements are true (and *understand* that there is nothing more to it). In particular, WM has no clue when he insists that that proves the existence of dark numbers. (Not that he has a clue in pretty much anything else, either.)

On Sunday, 17 October 2021 at 07:18:56 UTC-3, WM wrote:
> FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 20:50:14 UTC+2:
> > WM laid this down on his screen :
> > >>> there are not more than four endsegments containing the natural
> > >>> numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
> > I understand *and* disagree. I am not alone in this.
> Really? Who else beliefs that there are more than four endsegments?

You think this falsification is cute, don't you? You took the quotation out of context, you *FUCKING LIAR*.

On Sunday, 17 October 2021 at 09:57:03 UTC-3, WM wrote:
[...]
> Each definable natural number has infinitely many successors which are also natural numbers. So the infinite set |N of natural numbers is complete.

Correct.

> Each infinite endsegment has not infinitely many succeeding infinite endsegments because the infinite contents cannot be used as indexes simultaneously.

*BULLSHIT*! This nonsense seems to stem again from your utter inability to respect the order of quantifiers.

Gus Gassmann wrote:
> On Sunday, 17 October 2021 at 07:18:56 UTC-3, WM wrote:
>> FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 20:50:14 UTC+2:
>>> WM laid this down on his screen :
>>>>>> there are not more than four endsegments containing the natural
>>>>>> numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
>>> I understand *and* disagree. I am not alone in this.
>> Really? Who else beliefs that there are more than four endsegments?
>
> You think this falsification is cute, don't you? You took the quotation out of context, you *FUCKING LIAR*.

Crank Wolfgang Mueckenheim, from Hochschule Augsburg, is a charming
mix of dishonesty and stupidity. How come Germany allow such a
disgraceful fraud to "teach"???

FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 15:17:22 UTC+2:
> WM formulated on Sunday :
> > FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 14:44:19 UTC+2:
> >> WM wrote :
> >
> >>> (C) There are not more than finitely many infinite endsegments.
> >> A declaration with nothing to back it up. Each natural number has such
> >> a following, no exceptions.
> >
> > Each definable natural number
> Which one isn't?

Almost all needed to fill the set ℕ and to get |ℕ \ {1, 2, 3, ...}| = 0 in spite of ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

Regards, WM

zelos...@gmail.com schrieb am Sonntag, 17. Oktober 2021 um 15:14:44 UTC+2:
> söndag 17 oktober 2021 kl. 15:00:10 UTC+2 skrev WM:
> > FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 14:53:42 UTC+2:
> > > WM formulated on Sunday :
> > > > FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 20:50:14 UTC+2:
> > > >> WM laid this down on his screen :
> > > >
> > > >>>>> there are not more than four endsegments containing the natural
> > > >>>>> numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
> > > >
> > > >> I understand *and* disagree. I am not alone in this.
> > > >
> > > > Really? Who else beliefs that there are more than four endsegments?
> > > Nice editing job. You know of course that I was disagreeing with your
> > > "definable" and "dark" weasel-word numbers in the rest of your post.
> > No, how should I? If you understand that above only four can exist, then you could also understand that only n can exist containing {n, n+1, n+2, ....}. Then you could also understand that this is true for every n starting an infinite set. What is the difference?

> Yes, it is weasel words

No, every infinite endsegment of ℕ starts at some finite natural number.

Regards, WM

After serious thinking Gus Gassmann wrote :
> On Saturday, 16 October 2021 at 15:50:14 UTC-3, FromTheRafters wrote:
> [...]
>>> Fact is: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo and |ℕ \ {1, 2, 3, ...}|
>>> = 0. Whether you agree and understand or disagree and not understand is
>>> irrelevant.
>> I understand *and* disagree. I am not alone in this.
>
> Wait. I agree with these statements. What I (and I hope, you) disagree with
> is that this has any significance beyond the rather trivial fact that both
> statements are true (and *understand* that there is nothing more to it). In
> particular, WM has no clue when he insists that that proves the existence of
> dark numbers. (Not that he has a clue in pretty much anything else, either.)

I disagree because 'defined natural numbers' has not been adequately
defined. He implies that they are a finite (but growing) proper subset
of the naturals but fails to convince anyone of that and that his dark
numbers are needed to 'complete' the set of natural numbers.

Gus Gassmann schrieb am Sonntag, 17. Oktober 2021 um 17:47:34 UTC+2:
> On Sunday, 17 October 2021 at 09:57:03 UTC-3, WM wrote:
> [...]
> > Each definable natural number has infinitely many successors which are also natural numbers. So the infinite set |N of natural numbers is complete..
> Correct.
> > Each infinite endsegment has not infinitely many succeeding infinite endsegments because the infinite contents cannot be used as indexes simultaneously.
> This nonsense seems to stem again from your utter inability to respect the order of quantifiers.

This simple fact cannot be remedied by quantifier tricks. Infinite endsegments have an infinite intersection and there are not aleph_0 infinite endsegments. There are not aleph_0 natural numbers having alweh_0 successors.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

Regards, WM

FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 20:33:53 UTC+2:

> I disagree because 'defined natural numbers' has not been adequately
> defined.

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

Regards, WM

WM laid this down on his screen :
> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 15:17:22 UTC+2:
>> WM formulated on Sunday :
>>> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 14:44:19 UTC+2:
>>>> WM wrote :
>>>
>>>>> (C) There are not more than finitely many infinite endsegments.
>>>> A declaration with nothing to back it up. Each natural number has such
>>>> a following, no exceptions.
>>>
>>> Each definable natural number
>> Which one isn't?
>
> Almost all needed to fill the set ℕ and to get |ℕ \ {1, 2, 3, ...}| = 0 in
> spite of ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

It cannot be done. There is no zero in your set of naturals. Get with
the program and adjoin zero. Then, you can have a FISON set like {0, 1,
2, 3, 4, 5, 6, 7, 8, 9} and have its cardinality be your associated
endsegment's index. That way you avoid the mental torture of
'exhausting' your supply of natural numbers.

On Sunday, October 17, 2021 at 6:23:28 AM UTC-4, WM wrote:
> Dan Christensen schrieb am Samstag, 16. Oktober 2021 um 19:37:03 UTC+2:
> > On Saturday, October 16, 2021 at 12:21:12 PM UTC-4, WM wrote:
> > > FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 13:00:37 UTC+2:
> >
> > > > > Note that between every definable natnumber and ω there are infinitely many
> > > > > dark natnumbers ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > > > > while between all natnumbers and ω there is nothing
> > > > >> |ℕ \ {1, 2, 3, ...}| = 0 .
> > > > Wrong, dark numbers don't exist.
> > > Fact is: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ...., n}| = ℵo and |ℕ \ {1, 2, 3, ...}| = 0. Whether you agree and understand or disagree and not understand is irrelevant.
> > >
> > But wait! You recently wrote:
> >
> > "ℕ_def = {k ∈ ℕ : |∩{E(1), E(2), ..., E(k)}|= ℵo}"
> > --sci.math "ZFC knows potential infinity," 2021-08-11
> >
> > Und so, N_def = N.
> No, that is a erroneous conclusion.

Wrong again, Mucke. Recall that:

E(n) = {n, n+1, n+2, ... } for all n in N.

|E(n)|= ℵo for all n in N

∩{E(1), E(2), ..., E(k)} = E(k) for all k in N

|∩{E(1), E(2), ..., E(k)}| = |E(k)| = ℵo for all k in N..

Substituting, N_def = {k ∈ ℕ : |∩{E(1), E(2), ..., E(k)}|= ℵo} = {k ∈ ℕ : |E(k)}|= ℵo} = {k ∈ ℕ : ℵo = ℵo} = N.

Don't you feel silly? After all those years, you are reduced to this. All that wasted time and effort for NOTHING! Very sad indeed. Time to cut your losses and move on, Mucke.

Dan

WM explained :
> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 20:33:53 UTC+2:
>
>> I disagree because 'defined natural numbers' has not been adequately
>> defined.
>
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> contrary to
> ℕ \ {1, 2, 3, ...} = { }

Are you claiming that as a definition?

FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 21:18:23 UTC+2:
> WM explained :
> > FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 20:33:53 UTC+2:
> >
> >> I disagree because 'defined natural numbers' has not been adequately
> >> defined.
> >
> > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > contrary to
> > ℕ \ {1, 2, 3, ...} = { }
> Are you claiming that as a definition?

Definition: A natural number is "identified" or (individually) "defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0. All other natural numbers are called dark natural numbers.
Communication can occur
 by direct description in the unary system like ||||||| or as many beeps, flashes, or raps,
 by a finite initial segment of natural numbers (1, 2, 3, 4, 5, 6, 7) called a FISON,
 as n-ary representation, for instance binary 111 or decimal 7,
 by indirect description like "the number of colours of the rainbow",
 by other words known to sender and receiver like "seven".
https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 212.

Regards, WM

FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 21:13:12 UTC+2:
> WM laid this down on his screen :
> > FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 15:17:22 UTC+2:
> >> WM formulated on Sunday :
> >>> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 14:44:19 UTC+2:
> >>>> WM wrote :
> >>>
> >>>>> (C) There are not more than finitely many infinite endsegments.
> >>>> A declaration with nothing to back it up. Each natural number has such
> >>>> a following, no exceptions.
> >>>
> >>> Each definable natural number
> >> Which one isn't?
> >
> > Almost all needed to fill the set ℕ and to get |ℕ \ {1, 2, 3, ...}| = 0 in
> > spite of ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> It cannot be done. There is no zero in your set of naturals.

No problem.

Regards, WM

Dan Christensen schrieb am Sonntag, 17. Oktober 2021 um 21:13:23 UTC+2:
> On Sunday, October 17, 2021 at 6:23:28 AM UTC-4, WM wrote:

> > > > Fact is: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo and |ℕ \ {1, 2, 3, ...}| = 0. Whether you agree and understand or disagree and not understand is irrelevant.
> > > >
> > > But wait! You recently wrote:
> > >
> > > "ℕ_def = {k ∈ ℕ : |∩{E(1), E(2), ...., E(k)}|= ℵo}"
> > > --sci.math "ZFC knows potential infinity," 2021-08-11
> > >
> > > Und so, N_def = N.
> > No, that is an erroneous conclusion.
>
> E(n) = {n, n+1, n+2, ... } for all n in N.
>
> |E(n)|= ℵo for all n in N

That appears so to the superficial thinker.
(A) There are not more than four endsegments containing the naturalnumbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
(B) There are not more than n endsegments containing the natnumbers {n, n+1, n+2, ...}.
(C) There are not more than finitely many infinite endsegments.

Regards, WM

On Sunday, 17 October 2021 at 15:51:47 UTC-3, WM wrote:
> Gus Gassmann schrieb am Sonntag, 17. Oktober 2021 um 17:47:34 UTC+2:
> > On Sunday, 17 October 2021 at 09:57:03 UTC-3, WM wrote:
> > [...]
> > > Each definable natural number has infinitely many successors which are also natural numbers. So the infinite set |N of natural numbers is complete.
> > Correct.
> > > Each infinite endsegment has not infinitely many succeeding infinite endsegments because the infinite contents cannot be used as indexes simultaneously.
> > This nonsense seems to stem again from your utter inability to respect the order of quantifiers.
> This simple fact cannot be remedied by quantifier tricks. Infinite endsegments have an infinite intersection and there are not aleph_0 infinite endsegments. There are not aleph_0 natural numbers having alweh_0 successors.

Of course there are! *EVERY* one of the aleph_0 natural numbers has aleph_0 successors. The fact that you have no way of processing that points to a deep mental defect on your part, not any deficiencies in Cantor's work.

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

Finite intersections of end segments (as you also wrote!) have cardinality aleph_0. To extend that to infinite intersections you are switching quantifiers. This has been pointed out to you a good many times, but you are utterly incapable of understanding. That is *YOUR* mental defect.

On Sunday, 17 October 2021 at 16:13:12 UTC-3, FromTheRafters wrote:
> WM laid this down on his screen :
> > FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 15:17:22 UTC+2:
> >> WM formulated on Sunday :
> >>> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 14:44:19 UTC+2:
> >>>> WM wrote :
> >>>
> >>>>> (C) There are not more than finitely many infinite endsegments.
> >>>> A declaration with nothing to back it up. Each natural number has such
> >>>> a following, no exceptions.
> >>>
> >>> Each definable natural number
> >> Which one isn't?
> >
> > Almost all needed to fill the set ℕ and to get |ℕ \ {1, 2, 3, ...}| = 0 in
> > spite of ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> It cannot be done. There is no zero in your set of naturals. Get with
> the program and adjoin zero. Then, you can have a FISON set like {0, 1,
> 2, 3, 4, 5, 6, 7, 8, 9} and have its cardinality be your associated
> endsegment's index. That way you avoid the mental torture of
> 'exhausting' your supply of natural numbers.

Another easy remedy would be defining the end segments by E(n) = {n+1, n+2, n+3, ... } as even the great professor once stated, while even admitting he wished he had used that definition from the get-go. Seems the entire sordid affair and his (pretended?) mis-understanding are altogether self-inflicted. (Of course, in his current mental state I am not sure he remembers that particular episode.)

On Sunday, 17 October 2021 at 16:49:43 UTC-3, WM wrote:
> Dan Christensen schrieb am Sonntag, 17. Oktober 2021 um 21:13:23 UTC+2:
> > On Sunday, October 17, 2021 at 6:23:28 AM UTC-4, WM wrote:
>
> > > > > Fact is: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo and |ℕ \ {1, 2, 3, ...}| = 0. Whether you agree and understand or disagree and not understand is irrelevant.
> > > > >
> > > > But wait! You recently wrote:
> > > >
> > > > "ℕ_def = {k ∈ ℕ : |∩{E(1), E(2), ...., E(k)}|= ℵo}"
> > > > --sci.math "ZFC knows potential infinity," 2021-08-11
> > > >
> > > > Und so, N_def = N.
> > > No, that is an erroneous conclusion.
> >
> > E(n) = {n, n+1, n+2, ... } for all n in N.
> >
> > |E(n)|= ℵo for all n in N
> That appears so to the superficial thinker.

For anyone who is not a brain-dead pretend-mathematician, it is an easy consequence of the principle of induction.

> (A) There are not more than four endsegments containing the naturalnumbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
> (B) There are not more than n endsegments containing the natnumbers {n, n+1, n+2, ...}.
> (C) There are not more than finitely many infinite endsegments.

(C), of course, does not follow. Only a brain-dead pretend-mathematician can even think it might follow.

On Sunday, October 17, 2021 at 3:49:43 PM UTC-4, WM wrote:
> Dan Christensen schrieb am Sonntag, 17. Oktober 2021 um 21:13:23 UTC+2:
> > On Sunday, October 17, 2021 at 6:23:28 AM UTC-4, WM wrote:
>
> > > > > Fact is: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo and |ℕ \ {1, 2, 3, ...}| = 0. Whether you agree and understand or disagree and not understand is irrelevant.
> > > > >
> > > > But wait! You recently wrote:
> > > >
> > > > "ℕ_def = {k ∈ ℕ : |∩{E(1), E(2), ...., E(k)}|= ℵo}"
> > > > --sci.math "ZFC knows potential infinity," 2021-08-11
> > > >
> > > > Und so, N_def = N.
> > > No, that is an erroneous conclusion.
> >
> > E(n) = {n, n+1, n+2, ... } for all n in N.
> >
> > |E(n)|= ℵo for all n in N

> That appears so to the superficial thinker.

Still in denial, I see. You are looking like a superficial idiot here, Mucke.

> (A) There are not more than four endsegments containing the naturalnumbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.

OK, assuming N = {1, 2, 3, 4, ... }
> (B) There are not more than n endsegments containing the natnumbers {n, n+1, n+2, ...}.

Actually, there are precisely n such end-segments.

> (C) There are not more than finitely many infinite endsegments.
>

Jeez yer dumb! I will add that to your list idiocies.

E(1) = {1, 2, 3, 4, ... }
E(2) = {2, 3, 4, 5, ... }
E(3) = {3, 4, 5, 6, ... }
....
E(n) = {n, n+1, n+2, ... }
....

Contrary to your outrageous claim here, each of these infinitely many end-segments has in it infinitely many elements.

If you leave quietly, we won't tell anyone.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

It happens that WM formulated :
> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 21:18:23 UTC+2:
>> WM explained :
>>> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 20:33:53 UTC+2:
>>>
>>>> I disagree because 'defined natural numbers' has not been adequately
>>>> defined.
>>>
>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>> contrary to
>>> ℕ \ {1, 2, 3, ...} = { }
>> Are you claiming that as a definition?
>
> Definition: A natural number is "identified" or (individually) "defined" or
> "instantiated" if it can be communicated such that sender and receiver
> understand the same and can link it by a finite initial segment to the origin
> 0. All other natural numbers are called dark natural numbers. Communication
> can occur  by direct description in the unary system like ||||||| or as many
> beeps, flashes, or raps,  by a finite initial segment of natural numbers (1,
> 2, 3, 4, 5, 6, 7) called a FISON,  as n-ary representation, for instance
> binary 111 or decimal 7,  by indirect description like "the number of
> colours of the rainbow",  by other words known to sender and receiver like
> "seven". https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p.
> 212.
>
> Regards, WM

Thanks, I needed a good laugh.

On Sunday, October 17, 2021 at 3:43:27 PM UTC-4, WM wrote:
> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 21:18:23 UTC+2:
> > WM explained :
> > > FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 20:33:53 UTC+2:
> > >
> > >> I disagree because 'defined natural numbers' has not been adequately
> > >> defined.
> > >
> > > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > > contrary to
> > > ℕ \ {1, 2, 3, ...} = { }
> > Are you claiming that as a definition?
> Definition: A natural number is "identified" or (individually) "defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0. All other natural numbers are called dark natural numbers.
> Communication can occur
>  by direct description in the unary system like ||||||| or as many beeps, flashes, or raps,
>  by a finite initial segment of natural numbers (1, 2, 3, 4, 5, 6, 7) called a FISON,
>  as n-ary representation, for instance binary 111 or decimal 7,
>  by indirect description like "the number of colours of the rainbow",
>  by other words known to sender and receiver like "seven".
> https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 212.
>

Don't waste your time, Mucke. I would stick to Peano's Axioms and set theory. We know they work. Your goofy little system, not so much. Heck, today you proclaimed, "There are not more than finitely many infinite endsegments [in the set of natural numbers]." Sounds like something AP would say. It doesn't exactly inspire confidence.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

On 10/17/2021 2:43 PM, WM wrote:
> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 21:18:23 UTC+2:
>> WM explained :
>>> FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 20:33:53 UTC+2:
>>>
>>>> I disagree because 'defined natural numbers' has not been adequately
>>>> defined.
>>>
>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>> contrary to
>>> ℕ \ {1, 2, 3, ...} = { }
>> Are you claiming that as a definition?
>
> Definition: A natural number is "identified" or (individually) "defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0. All other natural numbers are called dark natural numbers.
> Communication can occur
>  by direct description in the unary system like ||||||| or as many beeps, flashes, or raps,
>  by a finite initial segment of natural numbers (1, 2, 3, 4, 5, 6, 7) called a FISON,
>  as n-ary representation, for instance binary 111 or decimal 7,
>  by indirect description like "the number of colours of the rainbow",
>  by other words known to sender and receiver like "seven".
> https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 212.
>
> Regards, WM
>

THat is a Daffynition! totally unworkable pile of steaming poo.

Beeps and raps, and flashing lights

no reqirement to write anything down at all