zelos...@gmail.com schrieb am Dienstag, 20. September 2022 um 13:42:24 UTC+2:
> måndag 19 september 2022 kl. 19:12:02 UTC+2 skrev WM:
> > Fritz Feldhase schrieb am Montag, 19. September 2022 um 14:03:10 UTC+2:
> > > On Sunday, September 18, 2022 at 8:44:37 PM UTC+2, WM wrote:
> > >
> > > > Every n can be reached by the process 1, 2, 3, ..., n.
> > > 1, 2, 3, ..., n.
> > > is not a "process",
> > It is a finite process while 1, 2, 3, ... is an infinite process.
> >
> > Look, Cantor talks about processes, for instance here: ein drittes Prinzip, welches ich das Hemmungs- oder Beschränkungsprinzip nenne, entgegen, wodurch dem durchaus endlosen Bildungsprozeß sukzessive gewisse Schranken auferlegt werden, [Cantor]
> > and here: Es wird daher zunächst der Anschein erweckt, als ob wir uns bei dieser Bildungsweise neuer ganzer bestimmt-unendlicher Zahlen ins Grenzenlose hin verlieren müßten, und daß wir außerstande seien, diesem endlosen Prozeß einen gewissen vorläufigen Abschluß zu geben, [Cantor]
> >
> > Everybody can see that you are wrong. Can you see it too?

> > To go through a sequence means a process.
> No it doesn't because mathematics DO NOT DO PROCESSES!

Cantor's mathematics does. See above. If you can't read or can't understand, then keep out of this discussion pleeze.
> > >
> > > It seems that you just have discovered the fact that for any natural number n there are n - 1 smaller natural numbers
> > I have discovered that for dark numbers this is not the case. Simplest proof: Going from omega to one is aprocess passing only finitely many numbers.
> No, you have ASSERTED it for your MADE UP numbers that you CANNOT prove EXISTS!

Proof is this:

k = (m + n - 1)(m + n - 2)/2 + m yields the sequence of indexed fractions:
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
....

Integer fractions are denoted by X's:

XOOO...
XOOO...
XOOO...
XOOO...
....

In the end, when all exchanges of X and O have been carried through according to Cantor we have

XXXX...
XXXX...
XXXX...
XXXX...
....

but no O has left the matrix. Where are they? Dark.

Regards, WM

zelos...@gmail.com schrieb am Dienstag, 20. September 2022 um 13:41:35 UTC+2:

> Inducion is true and works in cantors as well because it makes no DIFFERENCE there!

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

Regards, WM

Jim Burns schrieb am Montag, 19. September 2022 um 21:01:20 UTC+2:
>
> There are two concepts in play here.

Very well observed!

> | You can't get there from here.

Actual infinity.

> and
> | There is no there.

Potential infinity.
>
> Induction reasons about all the places
> where you _can_ get from here.

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

> For each natural number,
> for each split of that number's precursors,
> we _can_ get there from here.

For each definable number, to be precise. In potential infinity there are only definable numbers. But in actual infinity, there is more.

Regards, WM

Python schrieb am Sonntag, 18. September 2022 um 23:53:55 UTC+2:
> WM wrote:
> > Ben Bacarisse schrieb am Sonntag, 18. September 2022 um 22:04:56 UTC+2:
> ...
> >> You've had many opportunities to revise it.
> >
> > No, there will be no further printed edition.
> That is a very good news.

I am glad that I could please you. Alas PDFs will continue to be sold.

Regards, WM

On 9/20/2022 9:01 AM, WM wrote:
> zelos...@gmail.com schrieb am Dienstag, 20. September 2022 um 13:42:24 UTC+2:
>> måndag 19 september 2022 kl. 19:12:02 UTC+2 skrev WM:
>>> Fritz Feldhase schrieb am Montag, 19. September 2022 um 14:03:10 UTC+2:
>>>> On Sunday, September 18, 2022 at 8:44:37 PM UTC+2, WM wrote:
>>>>
>>>>> Every n can be reached by the process 1, 2, 3, ..., n.
>>>> 1, 2, 3, ..., n.
>>>> is not a "process",
>>> It is a finite process while 1, 2, 3, ... is an infinite process.
>>>
>>> Look, Cantor talks about processes, for instance here: ein drittes Prinzip, welches ich das Hemmungs- oder Beschränkungsprinzip nenne, entgegen, wodurch dem durchaus endlosen Bildungsprozeß sukzessive gewisse Schranken auferlegt werden, [Cantor]
>>> and here: Es wird daher zunächst der Anschein erweckt, als ob wir uns bei dieser Bildungsweise neuer ganzer bestimmt-unendlicher Zahlen ins Grenzenlose hin verlieren müßten, und daß wir außerstande seien, diesem endlosen Prozeß einen gewissen vorläufigen Abschluß zu geben, [Cantor]
>>>
>>> Everybody can see that you are wrong. Can you see it too?
>
>>> To go through a sequence means a process.
>> No it doesn't because mathematics DO NOT DO PROCESSES!
>
> Cantor's mathematics does. See above. If you can't read or can't understand, then keep out of this discussion pleeze.
>>>>
>>>> It seems that you just have discovered the fact that for any natural number n there are n - 1 smaller natural numbers
>>> I have discovered that for dark numbers this is not the case. Simplest proof: Going from omega to one is aprocess passing only finitely many numbers.
>> No, you have ASSERTED it for your MADE UP numbers that you CANNOT prove EXISTS!
>
> Proof is this:
>
> k = (m + n - 1)(m + n - 2)/2 + m yields the sequence of indexed fractions:
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
>
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
>
> Integer fractions are denoted by X's:
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ...
>
> In the end, when all exchanges of X and O have been carried through according to Cantor we have
>
> XXXX...
> XXXX...
> XXXX...
> XXXX...
> ...
>
> but no O has left the matrix.

Liar, you replaced each O with an X when YOU indexed it.

>
> Regards, WM

Sergi o schrieb am Dienstag, 20. September 2022 um 16:50:25 UTC+2:

> you replaced each O with an X when YOU indexed it.
The O moved to the position where I took the index from.

Regards, WM

On 9/20/2022 10:09 AM, WM wrote:
> Sergi o schrieb am Dienstag, 20. September 2022 um 16:50:25 UTC+2:
>
>> you replaced each O with an X when YOU indexed it.
>
> The O moved to the position where I took the index from.
>
> Regards, WM

no Deceiver,

you peeled off the "O pastie" on that fraction when you indexed it, replacing that "O pastie" with an "X pastie".

no O's were removed, they were replaced by X.

Your question of where did all the O's go, is over ridden by where did all the X's come from!

and keep your pasties to yourself!

On 9/20/2022 9:03 AM, WM wrote:
> zelos...@gmail.com schrieb am Dienstag, 20. September 2022 um 13:41:35 UTC+2:
>
>> Inducion is true and works in cantors as well because it makes no DIFFERENCE there!
>
> Induction proves: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

Wrong. Induction is not involved at all.

When you subtract a finite set from an infinite set, you still have an infinite set.

_def is troll food.

>
> Regards, WM

On Sunday, September 18, 2022 at 10:52:08 PM UTC+2, WM wrote:
> Ben Bacarisse schrieb am Sonntag, 18. September 2022 um 22:04:56 UTC+2:
> > WM <askas...@gmail.com> writes:
> > (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> > Unendlichen" at Hochschule Augsburg.)
> > > Ben Bacarisse schrieb am Freitag, 16. September 2022 um 16:42:09 UTC+2:
> > >
> > >> WM likes to talk about processes because they don't end, but properties
> > >> of functions can be proved using finite arguments. In effect he wants
> > >> to deny another part of his book -- the final words in a proof by
> > >> induction. He'd be quite happy with "P(1) and P(x) implies P(x+1)" to
> > >> be the end of it, but because he needs to sell the book, he concludes
> > >> (as does every author)
> > >>
> > >> "so ist sie für alle natürlichen Zahlen richtig"
> > >
> > > Yes, alas only for definable natural numbers.
> > So another mistake in your book! How did this mistake stay in the book
> > for so long?
> Not a mistake. In potential infinity all definable natnumbers all all natnumbers. I told you frequently already: Only if accepting Cantor, dark numbers come into the play. Nobody is forced to accept Cantor.
> > You've had many opportunities to revise it.
> No, there will be no further printed edition. The seventh edition of my Geschichte des Unendlichen and the fourth edition of Mathematik für die ersten Semester will remain the final editions because any number of copies can be made by PDF.
> >

You seem to have misunderstood something: PDF's are editable at will.
Printed books are not.
But I understand the problem: You can't handle the resulting page layout
shift in Tex, so you would have to hire somebody for that; and the publisher
would also charge you a fee.
Therefore, your masterwork will have to remain imperfect, even the embarrassing
arithmetic of the divergent integrals will be there forever for all to see.

On Tuesday, September 20, 2022 at 10:08:05 AM UTC-4, WM wrote:
> Jim Burns schrieb am Montag, 19. September 2022 um 21:01:20 UTC+2:
> >
> > There are two concepts in play here.
> Very well observed!
> > | You can't get there from here.
> Actual infinity.
> > and
> > | There is no there.
> Potential infinity.
> >
> > Induction reasons about all the places
> > where you _can_ get from here.
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > For each natural number,
> > for each split of that number's precursors,
> > we _can_ get there from here.
> For each definable number, to be precise. In potential infinity there are only definable numbers. But in actual infinity, there is more.
>
> Regards, WM

I'm afraid I do see WM's argument here.
I suppose anywhere that Jim uses '_word_' he means to say that the language is imperfect.
As the word _can_ is conceded as a weak form then too much wiggle room is allowed.
I believe we are all guilty somehow of practicing too loosely.
I do think that my own stress on the ellipses and their connection to induction is relevant.
In defense of my new infinite values: they are literally numerical constructions or representations.
In this regard they are secured by the freedom to construct.
Their weaknesses should be exposed in their ambiguities.
That this sort of language is their failing point is roughly the failing point of all discussions of infinity.
Drawing the line and blacking out everything beyond it then things go dark.
As to how much is beyond that line versus back from it: here I draw that line at the first usage of ellipses, which are the first usage of iteration, which we naturally couple into induction. Every usage of the ellipses _can_ go on from there (if they are well formed), but none of them _do_ go on from there.

As I review the Peano axioms there is very little there. By their own abstraction and lack of regard for digits they will never make it past nine. What allows for the interpretation of sequences of digits as large numbers? None of these mechanics of number are present in the Peano axioms, and so they are incomplete. As a definition of a number system they will not allow for the usage of 123 as a single value. As a primitive rendition of number since no modulo mechanics are mentioned then all who invoke modulo mechanics are not practicing Peano Arithmetic. This is how fundamental the description is. To admit that the Peano Axioms are not complete and that there is this entire other layer to number and particularly to large number really makes them look rather flat and drab. Peano numbers lack the whir and the click and the pop of our ordinary base ten numbers. They lack the ambiguities, too. The magical 'successor' is just an avoidance mechanism in some regards.

Convention and notation are valid areas to review here. That numerical representation is completely avoided by these types who think that they are doing some superior work, while they will never make it past nine afaict is problematic.

I guess I'll try this method here to counter Jim's split argument:

34 is a natural number.
334 is a natural number.
3334 is a natural number.
Therefor 333...3334 is a natural number by induction.

I guess these four lines above should be taken as a free-standing argument, without worrying about any of the background information.
Induction can be this simple. I suppose that for somebody to disagree they will have to deny that this is a valid usage of induction.
As we come to developing the product on such values it becomes apparent that this inductive form is totally relevant. That here we are leveraging the modulo-ten form of number and that this view of number is not on the radar of ordinary natural analysis is fairly apparent.

On 9/20/2022 11:21 AM, Timothy Golden wrote:
> On Tuesday, September 20, 2022 at 10:08:05 AM UTC-4, WM wrote:
>> Jim Burns schrieb am Montag, 19. September 2022 um 21:01:20 UTC+2:
>>>
>>> There are two concepts in play here.
>> Very well observed!
>>> | You can't get there from here.
>> Actual infinity.
>>> and
>>> | There is no there.
>> Potential infinity.
>>>
>>> Induction reasons about all the places
>>> where you _can_ get from here.
>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>> For each natural number,
>>> for each split of that number's precursors,
>>> we _can_ get there from here.
>> For each definable number, to be precise. In potential infinity there are only definable numbers. But in actual infinity, there is more.
>>
>> Regards, WM
>
> I'm afraid I do see WM's argument here.
> I suppose anywhere that Jim uses '_word_' he means to say that the language is imperfect.
> As the word _can_ is conceded as a weak form then too much wiggle room is allowed.
> I believe we are all guilty somehow of practicing too loosely.
> I do think that my own stress on the ellipses and their connection to induction is relevant.
> In defense of my new infinite values: they are literally numerical constructions or representations.
> In this regard they are secured by the freedom to construct.
> Their weaknesses should be exposed in their ambiguities.
> That this sort of language is their failing point is roughly the failing point of all discussions of infinity.
> Drawing the line and blacking out everything beyond it then things go dark.
> As to how much is beyond that line versus back from it: here I draw that line at the first usage of ellipses, which are the first usage of iteration, which we naturally couple into induction. Every usage of the ellipses _can_ go on from there (if they are well formed), but none of them _do_ go on from there.
>
> As I review the Peano axioms there is very little there. By their own abstraction and lack of regard for digits they will never make it past nine. What allows for the interpretation of sequences of digits as large numbers? None of these mechanics of number are present in the Peano axioms, and so they are incomplete. As a definition of a number system they will not allow for the usage of 123 as a single value. As a primitive rendition of number since no modulo mechanics are mentioned then all who invoke modulo mechanics are not practicing Peano Arithmetic. This is how fundamental the description is. To admit that the Peano Axioms are not complete and that there is this entire other layer to number and particularly to large number really makes them look rather flat and drab. Peano numbers lack the whir and the click and the pop of our ordinary base ten numbers. They lack the ambiguities, too. The magical 'successor' is just an avoidance mechanism in some regards.
>
> Convention and notation are valid areas to review here. That numerical representation is completely avoided by these types who think that they are doing some superior work, while they will never make it past nine afaict is problematic.
>
> I guess I'll try this method here to counter Jim's split argument:
>
> 34 is a natural number.
> 334 is a natural number.
> 3334 is a natural number.
> Therefor 333...3334 is a natural number by induction.
>
> I guess these four lines above should be taken as a free-standing argument, without worrying about any of the background information.
> Induction can be this simple. I suppose that for somebody to disagree they will have to deny that this is a valid usage of induction.
> As we come to developing the product on such values it becomes apparent that this inductive form is totally relevant. That here we are leveraging the modulo-ten form of number and that this view of number is not on the radar of ordinary natural analysis is fairly apparent.

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On 9/20/2022 11:30 AM, Sergi o wrote:
> On 9/20/2022 11:21 AM, Timothy Golden wrote:
>> On Tuesday, September 20, 2022 at 10:08:05 AM UTC-4, WM wrote:
>>> Jim Burns schrieb am Montag, 19. September 2022 um 21:01:20 UTC+2:
>>>>
>>>> There are two concepts in play here.
>>> Very well observed!
>>>> | You can't get there from here.
>>> Actual infinity.
>>>> and
>>>> | There is no there.
>>> Potential infinity.
>>>>
>>>> Induction reasons about all the places
>>>> where you _can_ get from here.
>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>>> For each natural number,
>>>> for each split of that number's precursors,
>>>> we _can_ get there from here.
>>> For each definable number, to be precise. In potential infinity there are only definable numbers. But in actual infinity, there is more.
>>>
>>> Regards, WM
>>
>ǝɹǝɥ ʇxǝʇ ɹǝ˙ʇuǝɹɐddɐ ʎlɹᴉɐɟ sᴉ sᴉsʎlɐuɐ lɐɹnʇɐu ʎɹɐuᴉpɹo ɟo ɹɐpɐɹ ǝɥʇ uo ʇou sᴉ ɹǝqɯnu ɟo ʍǝᴉʌ sᴉɥʇ ʇɐɥʇ puɐ ɹǝqɯnu ɟo ɯɹoɟ uǝʇ-olnpoɯ ǝɥʇ ƃuᴉƃɐɹǝʌǝl ǝɹɐ ǝʍ ǝɹǝɥ ʇɐɥ⊥ ˙ʇuɐʌǝlǝɹ ʎllɐʇoʇ sᴉ ɯɹoɟ ǝʌᴉʇɔnpuᴉ sᴉɥʇ ʇɐɥʇ ʇuǝɹɐddɐ sǝɯoɔǝq ʇᴉ sǝnlɐʌ ɥɔns uo ʇɔnpoɹd ǝɥʇ ƃuᴉdolǝʌǝp oʇ ǝɯoɔ ǝʍ s∀
˙uoᴉʇɔnpuᴉ ɟo ǝƃɐsn pᴉlɐʌ ɐ sᴉ sᴉɥʇ ʇɐɥʇ ʎuǝp oʇ ǝʌɐɥ llᴉʍ ʎǝɥʇ ǝǝɹƃɐsᴉp oʇ ʎpoqǝɯos ɹoɟ ʇɐɥʇ ǝsoddns I ˙ǝldɯᴉs sᴉɥʇ ǝq uɐɔ uoᴉʇɔnpuI
˙uoᴉʇɐɯɹoɟuᴉ punoɹƃʞɔɐq ǝɥʇ ɟo ʎuɐ ʇnoqɐ ƃuᴉʎɹɹoʍ ʇnoɥʇᴉʍ 'ʇuǝɯnƃɹɐ ƃuᴉpuɐʇs-ǝǝɹɟ ɐ sɐ uǝʞɐʇ ǝq plnoɥs ǝʌoqɐ sǝuᴉl ɹnoɟ ǝsǝɥʇ ssǝnƃ I

˙uoᴉʇɔnpuᴉ ʎq ɹǝqɯnu lɐɹnʇɐu ɐ sᴉ ㄣƐƐƐ˙˙˙ƐƐƐ ɹoɟǝɹǝɥ⊥
˙ɹǝqɯnu lɐɹnʇɐu ɐ sᴉ ㄣƐƐƐ
˙ɹǝqɯnu lɐɹnʇɐu ɐ sᴉ ㄣƐƐ
˙ɹǝqɯnu lɐɹnʇɐu ɐ sᴉ ㄣƐ

:ʇuǝɯnƃɹɐ ʇᴉlds s,ɯᴉſ ɹǝʇunoɔ oʇ ǝɹǝɥ poɥʇǝɯ sᴉɥʇ ʎɹʇ ll,I ssǝnƃ I

˙ɔᴉʇɐɯǝlqoɹd sᴉ ʇɔᴉɐɟɐ ǝuᴉu ʇsɐd ʇᴉ ǝʞɐɯ ɹǝʌǝu llᴉʍ ʎǝɥʇ ǝlᴉɥʍ 'ʞɹoʍ ɹoᴉɹǝdns ǝɯos ƃuᴉop ǝɹɐ ʎǝɥʇ ʇɐɥʇ ʞuᴉɥʇ oɥʍ sǝdʎʇ ǝsǝɥʇ ʎq pǝpᴉoʌɐ ʎlǝʇǝldɯoɔ sᴉ
uoᴉʇɐʇuǝsǝɹdǝɹ lɐɔᴉɹǝɯnu ʇɐɥ⊥ ˙ǝɹǝɥ ʍǝᴉʌǝɹ oʇ sɐǝɹɐ pᴉlɐʌ ǝɹɐ uoᴉʇɐʇou puɐ uoᴉʇuǝʌuoƆ

˙spɹɐƃǝɹ ǝɯos uᴉ ɯsᴉuɐɥɔǝɯ ǝɔuɐpᴉoʌɐ uɐ ʇsnɾ sᴉ ,ɹossǝɔɔns, lɐɔᴉƃɐɯ ǝɥ⊥ ˙ooʇ 'sǝᴉʇᴉnƃᴉqɯɐ ǝɥʇ ʞɔɐl ʎǝɥ⊥ ˙sɹǝqɯnu uǝʇ ǝsɐq ʎɹɐuᴉpɹo ɹno ɟo dod ǝɥʇ puɐ
ʞɔᴉlɔ ǝɥʇ puɐ ɹᴉɥʍ ǝɥʇ ʞɔɐl sɹǝqɯnu ouɐǝԀ ˙qɐɹp puɐ ʇɐlɟ ɹǝɥʇɐɹ ʞool ɯǝɥʇ sǝʞɐɯ ʎllɐǝɹ ɹǝqɯnu ǝƃɹɐl oʇ ʎlɹɐlnɔᴉʇɹɐd puɐ ɹǝqɯnu oʇ ɹǝʎɐl ɹǝɥʇo ǝɹᴉʇuǝ
sᴉɥʇ sᴉ ǝɹǝɥʇ ʇɐɥʇ puɐ ǝʇǝldɯoɔ ʇou ǝɹɐ sɯoᴉx∀ ouɐǝԀ ǝɥʇ ʇɐɥʇ ʇᴉɯpɐ o⊥ ˙sᴉ uoᴉʇdᴉɹɔsǝp ǝɥʇ lɐʇuǝɯɐpunɟ ʍoɥ sᴉ sᴉɥ⊥ ˙ɔᴉʇǝɯɥʇᴉɹ∀ ouɐǝԀ ƃuᴉɔᴉʇɔɐɹd ʇou ǝɹɐ
sɔᴉuɐɥɔǝɯ olnpoɯ ǝʞoʌuᴉ oɥʍ llɐ uǝɥʇ pǝuoᴉʇuǝɯ ǝɹɐ sɔᴉuɐɥɔǝɯ olnpoɯ ou ǝɔuᴉs ɹǝqɯnu ɟo uoᴉʇᴉpuǝɹ ǝʌᴉʇᴉɯᴉɹd ɐ s∀ ˙ǝnlɐʌ ǝlƃuᴉs ɐ sɐ ƐᄅƖ ɟo ǝƃɐsn ǝɥʇ ɹoɟ
ʍollɐ ʇou llᴉʍ ʎǝɥʇ ɯǝʇsʎs ɹǝqɯnu ɐ ɟo uoᴉʇᴉuᴉɟǝp ɐ s∀ ˙ǝʇǝldɯoɔuᴉ ǝɹɐ ʎǝɥʇ os puɐ 'sɯoᴉxɐ ouɐǝԀ ǝɥʇ uᴉ ʇuǝsǝɹd ǝɹɐ ɹǝqɯnu ɟo sɔᴉuɐɥɔǝɯ ǝsǝɥʇ ɟo ǝuoN
¿sɹǝqɯnu ǝƃɹɐl sɐ sʇᴉƃᴉp ɟo sǝɔuǝnbǝs ɟo uoᴉʇɐʇǝɹdɹǝʇuᴉ ǝɥʇ ɹoɟ sʍollɐ ʇɐɥM ˙ǝuᴉu ʇsɐd ʇᴉ ǝʞɐɯ ɹǝʌǝu llᴉʍ ʎǝɥʇ sʇᴉƃᴉp ɹoɟ pɹɐƃǝɹ ɟo ʞɔɐl puɐ uoᴉʇɔɐɹʇsqɐ
uʍo ɹᴉǝɥʇ ʎ𐐒 ˙ǝɹǝɥʇ ǝlʇʇᴉl ʎɹǝʌ sᴉ ǝɹǝɥʇ sɯoᴉxɐ ouɐǝԀ ǝɥʇ ʍǝᴉʌǝɹ I s∀ʇuƎ

On 9/20/2022 9:36 AM, Timothy Golden wrote:
> On Monday, September 19, 2022
> at 7:06:35 PM UTC-4, Jim Burns wrote:
>> On 9/19/2022 6:07 PM, Chris M. Thomasson wrote:
>>> On 9/19/2022 12:01 PM, Jim Burns wrote:

>>>> What does 333...34 mean?
>>>
>>> Something like the following expansion.
>>>
>>> 34
>>> 334
>>> 3334
>>> 33334
>>> 333334
>>> 3333334
>>> 33333334
>>> ...
>>>
>>> ;^)
>>
>> That's one of my not-making-sense guesses.
>>
>> If
>> 333...34 is after each of
>> 34
>> 334
>> 3334
>> 33334
>> 333334
>> 3333334
>> 33333334
>> ...
>> then
>> there is a split of all the precursors
>> for which each step
>> which starts before the split
>> ends before the split
>>
>> But then,
>> you can't get there from here.
>> No step exists
>> which starts before the split and
>> ends after the split.
>>
>> That meaning of 333...34 is one for which
>> you can't get there from here.
>
> Yet we just did get there from here.
> The representation is done.

I think you mean to say
there _is_ a there.

That's not the same as
you _can_ get there from here.

Getting there needs a step across each split.
Being doesn't need them.

| For each place which we can get to,
| the place one step further is also
| a place which we can get to.

Next, consider
the places we can get to from here, and
the places we can't get to from here.

Each place we can get to is before
each place we can't get to.
It's a split.
But it's a split without a step across.

A step across needs to begin on one side
and end on the other.
However,
| for each place which we can get to,
| the place one step further is also
| a place which we can get to.

For everything on the far side of
a split without a step,
you can't get there from here.

You can't get there from here,
but
you can describe them, and reason about them.
Which we do.

> I don't accept your criterion.

If
we only advance to successors, i to i+1,
and
there is a split such that
each near-side advance i,i+1
stays on the near side,
then
we do not advance to the far side.

That's my criterion for _places we can get to_

Suppose you don't accept it:
How do we advance to the far side?

On 9/20/2022 10:08 AM, WM wrote:
> Jim Burns schrieb am Montag,
> 19. September 2022 um 21:01:20 UTC+2:

>> There are two concepts in play here.
>
> Very well observed!
>
>> | You can't get there from here.
>
> Actual infinity.
>
>> and
>> | There is no there.
>
> Potential infinity.

What you (WM) mean by those terms
doesn't seem very stable.

For each place to which you can get,
there is one more step, and
that one more step takes you to
another place to which you can get.

Consider the split between
the places you get get from here
and
the places you can't get from here.

From any place on the near "can" side,
one more step takes you to
another place on the near "can" side.

Therefore,
no step takes you
from a place on the near "can" side
to a place on the far "can't" side.

Therefore,
you can't get there from here.

> In potential infinity
> there are only definable numbers.
> But in actual infinity,
> there is more.

If
you can get to p,
and
you can get to q,
then
you can get to s = p+q
and
you can get to k = (s-1)(s-2)/2+p

If
you can get to k
then
you can get to s = ⌈(((8k+1)¹ᐟ²+1)/2)⌉
and
you can get to p = k-(s-1)(s-2)/2
and
you can get to q = s-p

That is done without any use of _more_

>
>
>
>
>
>
> Regards, WM

Jim Burns schrieb am Dienstag, 20. September 2022 um 19:59:45 UTC+2:

> From any place on the near "can" side,
> one more step takes you to
> another place on the near "can" side.

Without end. Potential infinity.
>
> Therefore,
> no step takes you
> from a place on the near "can" side
> to a place on the far "can't" side.

So it is. The reason is the abyss of dark numbers between both sides.
>
> Therefore,
> you can't get there from here.

So it is. The reason is the abyss of dark numbers between both sides.

> > In potential infinity
> > there are only definable numbers.
> > But in actual infinity,
> > there is more.
> If
> you can get to p,
> and
> you can get to q,
> then
> you can get to s = p+q
> and
> you can get to k = (s-1)(s-2)/2+p
>
> If
> you can get to k
> then
> you can get to s = ⌈(((8k+1)¹ᐟ²+1)/2)⌉
> and
> you can get to p = k-(s-1)(s-2)/2
> and
> you can get to q = s-p
>
> That is done without any use of _more_

That is done without crossing the abyss of darkness. Not even ℵo/100000 is reached.

Regards, WM

On Tuesday, September 20, 2022 at 1:59:21 PM UTC-4, Jim Burns wrote:
> On 9/20/2022 9:36 AM, Timothy Golden wrote:
> > On Monday, September 19, 2022
> > at 7:06:35 PM UTC-4, Jim Burns wrote:
> >> On 9/19/2022 6:07 PM, Chris M. Thomasson wrote:
> >>> On 9/19/2022 12:01 PM, Jim Burns wrote:
>
> >>>> What does 333...34 mean?
> >>>
> >>> Something like the following expansion.
> >>>
> >>> 34
> >>> 334
> >>> 3334
> >>> 33334
> >>> 333334
> >>> 3333334
> >>> 33333334
> >>> ...
> >>>
> >>> ;^)
> >>
> >> That's one of my not-making-sense guesses.
> >>
> >> If
> >> 333...34 is after each of
> >> 34
> >> 334
> >> 3334
> >> 33334
> >> 333334
> >> 3333334
> >> 33333334
> >> ...
> >> then
> >> there is a split of all the precursors
> >> for which each step
> >> which starts before the split
> >> ends before the split
> >>
> >> But then,
> >> you can't get there from here.
> >> No step exists
> >> which starts before the split and
> >> ends after the split.
> >>
> >> That meaning of 333...34 is one for which
> >> you can't get there from here.
> >
> > Yet we just did get there from here.
> > The representation is done.
> I think you mean to say
> there _is_ a there.
> That's not the same as
> you _can_ get there from here.
>
> Getting there needs a step across each split.
> Being doesn't need them.
>
>
> | For each place which we can get to,
> | the place one step further is also
> | a place which we can get to.
>
> Next, consider
> the places we can get to from here, and
> the places we can't get to from here.
>
> Each place we can get to is before
> each place we can't get to.
> It's a split.
> But it's a split without a step across.
>
> A step across needs to begin on one side
> and end on the other.
> However,
> | for each place which we can get to,
> | the place one step further is also
> | a place which we can get to.
>
> For everything on the far side of
> a split without a step,
> you can't get there from here.
> You can't get there from here,
> but
> you can describe them, and reason about them.
> Which we do.
> > I don't accept your criterion.
> If
> we only advance to successors, i to i+1,
> and
> there is a split such that
> each near-side advance i,i+1
> stays on the near side,
> then
> we do not advance to the far side.
>
> That's my criterion for _places we can get to_
>
> Suppose you don't accept it:
> How do we advance to the far side?

I'm sorry. Maybe I should deny the split that you claim exists. Where exactly is that split? Why is it there?
I think it is appropriate that you have engaged a progression of digits.
What they expose is that, yes, you can keep on going.
There is no contradiction, and so induction holds up.
Possibly you do not accept the blurring of induction with ellipsis.
The far side is literally inductive in nature and hence the ellipsis.
At least this is my own choice of language.
I could see how somebody might try to draw a distinction, but I suspect it will only yield a subtle change.
Strange the wiki does not have a mathematical entry in the disambiguation: https://en.wikipedia.org/wiki/Ellipsis_(disambiguation)

And again, the sensibility of infinity from counting requires the ellipsis as well. The difference is that with the digits the accrual of value is exponential. The slow boat to infinity does seem to yield a different sense of it; almost as if you can't quite get there. These inductive values get there quickly and are clearly defined and constant. 333...34 is a fine instance of something so large and yet so well defined that there is no room to quibble over any of its digits. You really are dodging the heart of the matter, I think. As for splits, there is a need of one when we generate a product. Let's suppose that x is 333...32 and we'd like to consider 10x and whether 10x is greater than x. Without an aleph mark we'll be comparing 333...332 with 333...320, and it would appear at first glance as though we got something wrong. By introducing an aleph mark we can disambiguate:
x = | 333...32 .
10 x = 3 | 333...320
Particularly under product of two inductive values, which will require two sets of ellipses and two aleph marks when we are being fully careful. So you see, taking products of these does in fact lead to even larger numbers whose digits are still completely defined. I'm sorry to confuse your split with this aleph mark as they are two very different things; still this is as close as I can come to requiring anything special like that.

Oh: let's say you want to insert some commas every third digit, like is done with financial stuff. Would these be the sorts of splits you are talking about? That could be tricky up on the high side of the ellipses. I don't think these would work out so well. If you are tacking something like that onto this number system I'd say don't do it. Then again, you haven't really introduced any contradiction yet. I'm getting close, though.

I guess you are just horsing around. As I re-read your language:
"there is a split of all the precursors for which each step which starts before the split ends before the split"
it is totally meaningless. Why don't you go back there and introduce this 'split' more carefully. There is no problem.
I guess I used to take you more seriously.

On 9/20/2022 12:21 PM, Timothy Golden wrote:

> I suppose anywhere that Jim uses '_word_'
> he means to say that the language is imperfect.

No. Not here, at least.

Here, '_can_' is emphasis.
The distinction can/can't is critical, and
I don't want my reader (you or someone else)
to work any harder than necessary
to see what I'm saying.

My reason is NOT that I have a low opinion
of my reader's mental resources.
My reason is that I have a better use in mind
for my reader's mental resources.

( I assume those resources are finite,
( so whatever is used noticing 'can/can't'
( is NOT used elsewhere.
( My apologies if you are God.

Other places, I'll underline '_word_'
when I'm using it in a defining claim.
That's so that, when I use 'word' again,
my reader works less to find what I mean.

For example,
| In a _counting-order_
| for each split,
| there is a last-before and a first-after,
| and first-after is successor to last-before.

> The magical 'successor' is just
> an avoidance mechanism in some regards.

'Successor' is no more magical than
the rest of mathematics. Since I don't
know what "magic" is, I can't say more.

In my opinion, the apparent magic arises from
talking _only_ about things with a certain
description.

"Magically", we know something about each one
of (what sometimes is) infinitely-many --
even though we are finite, and it is literally
impossible for us to check them all, not even
in principle.

_How_ we know begins by knowing what we're
talking about.

If we're talking about _right triangles_
then we know that it's a three-sided polygon
with one of its angles being a right angle.

From that description of _it_ whatever _it_ is,
we can reason our way to knowing that
the square of _its_ longest side equals
the sum of the squares of _its_ two other sides.

"Magically", we know that about infinitely-many
right triangles. We _know_ it. We can never
check it for all right triangles, not even in
principle. Nonetheless, we _know_ it.

If we're talking about _natural numbers_
....

Ah, well. That's what this discussion is about,
isn't it? What is a natural number?

In order to pull the same finite-to-infinite
trick we pulled with right triangles,
it's not enough to say
| 0 is a natural number.
| 1 is a natural number.
| 2 is a natural number.
| 10^(10^(10^(10))) is a natural number.

We need a description of a natural number,
claims together true of a natural number
but not true together for anything else.

I claim that, for a natural number n,
you _can_ get there (n) from here (0),
by successor-steps i,i+1

That matches our more exalted uses of
'natural number' like the Peano axioms
and our less exalted uses like saying
how many sheep are in the pasture.

I expect agreement on what a natural number
is, but I can't force agreement.

However,
agreement is not so important as it might
seem at first.

Suppose you have something different that
you call "natural numbers". Then we are
talking about different things. That's all.

"Blue" sometimes refers to a color roughly
similar to that of a clear sky.
"Blue" sometimes refers to feeling sad.
Claims made with one use do not touch on
claims made with a different use.
As long as it's clear what's meant,
no harm, no foul.

> The magical 'successor' is just
> an avoidance mechanism in some regards.

What is often meant by "successor" is
the successor for von Neumann ordinals.
| y is the successor of x ==
| y = x ∪ {x}

I think that's delightful and elegant,
but it might not be your cup of tea.

There are also decimal numerals,
certain finite strings composed of
the characters 0 1 2 3 4 5 6 7 8 9

We can define a successor operation
for decimal numerals. Children learn to
do it when they're introduced to arithmetic.
It's called "counting".

As long as a successor operation has the
essential properties, the details of how
exactly the operation is implemented can be
swept under the rug.

Essential properties (for natural numbers):
| | If n and n-1 exists, then n+1 exists.
| | If m and n are different,
| then m+1 and n+1 are different.
| | If m and n are different,
| then m-1 and n-1 are different
| (if they exist).
| | 0 exists.
| 0+1 exists, = 1
| 0-1 dies not exist (for 'natural' purposes).

Probably your 333...34 has a successor which
obeys all that. That's not why 333...34 is
not a natural number.

A natural number is one of those
things-with-successor (like that) for which
you _can_ get there from here.

> I guess I'll try this method here
> to counter Jim's split argument:
> 34 is a natural number.
> 334 is a natural number.
> 3334 is a natural number.
> Therefor
> 333...3334 is a natural number by induction.

That's not induction.

What is induction?
See also upthread.

> I suppose that for somebody to disagree
> they will have to deny that
> this is a valid usage of induction.

Bingo. Not valid.

I think you are making having successors and
predecessors the requirement for induction.
They aren't the requirement.

It's a matter of being able to get there
from here ==
for each split of precursors, there is
the right kind of step across the split.

On 9/19/2022 7:13 PM, Sergi o wrote:
> On 9/19/2022 6:47 PM, Chris M. Thomasson wrote:
>> On 9/19/2022 4:42 PM, Chris M. Thomasson wrote:
>>> On 9/19/2022 3:51 PM, Sergi o wrote:
>>>> On 9/19/2022 5:07 PM, Chris M. Thomasson wrote:
>>>>> On 9/19/2022 12:01 PM, Jim Burns wrote:
>>>>>> On 9/19/2022 8:51 AM, Timothy Golden wrote:
>>>>>>> On Sunday, September 18, 2022
>>>>>>> at 3:36:24 PM UTC-4, Jim Burns wrote:
>>>>> [...]
>>>>>> I don't know what the ellipsis in 333...34
>>>>>> means. I've got a couple guesses, but they
>>>>>> don't make much sense.
>>>>>> And you are here to tell me what it means.
>>>>>> So, why guess?
>>>>>>
>>>>>> What does 333...34 mean?
>>>>> [...]
>>>>>
>>>>> Something like the following expansion.
>>>>>
>>>>> 34
>>>>> 334
>>>>> 3334
>>>>> 33334
>>>>> 333334
>>>>> 3333334
>>>>> 33333334
>>>>> ...
>>>>>
>>>>> ;^)
>>>>
>>>> apply a geometric recursive formula;
>>> [...]
>>>
>>> Okay, let me give it a go:
>>> _______________________________
>>> i[0] = 34
>>> i[n] = i[n-1] + 3*10^(n+1)
>>> _______________________________
>>>
>>> Therefore we get the following infinite expansion:
>>> _______________________________
>>> i[0] = 34
>>> i[1] = i[0] + 3*10^(1+1) = 34 + 300 = 334
>>> i[2] = i[1] + 3*10^(2+1) = 334 + 3000 = 3334
>>> i[3] = i[2] + 3*10^(3+1) = 3334 + 30000 = 33334
>>> ...
>>> _______________________________
>>>
>>> that is 333...34, Right?
>>
>> Oh, I see that you already have that except I started at index zero
>> instead of one, hence my 3*10^(n+1) addend to the previous result of
>> the recursion. I read "apply a geometric recursive formula" and just
>> starting writing out a recursive formula.
>
> you got it, simply  333...34 = i[k]  where k is the number of 3's
>
> [no more math in this one, except an algorithm to add the 9s and 0s into
> the geometric coefficients I derived, to get a general coef.   strange
> how that came out, but there is a simple solution somehow]
>
> a(6) = a(5) * 9.99998200003599992800014399712....
>
> next one predicted to be
>
> a(7) = a(6) * 9.999998200000359999928000014399712....
>
>        then   9.9999998200000035999999280000014399712....

Humm... I think I have a way to get a correct answer without using
recursion. In other words i[n+1]o does not need to depend on knowing
what i[n] is. A direct answer without need to depend on prior steps. I
might have some more time tonight to tinker with it. Have some ideas.

On 9/20/2022 2:32 PM, WM wrote:
> Jim Burns schrieb am Dienstag,
> 20. September 2022 um 19:59:45 UTC+2:

>> From any place on the near "can" side,
>> one more step takes you to
>> another place on the near "can" side.
>
> Without end. Potential infinity.

Virtus dormitiva.

>> Therefore,
>> no step takes you
>> from a place on the near "can" side
>> to a place on the far "can't" side.
>
> So it is.
> The reason is
> the abyss of dark numbers between both sides.

You quote the reason, just now.

A step which ends on the near side
can't end on the far side.

All steps end on the near side,
no steps end on the far side.

You can't get there from here.

Your dark numbers do not enter the picture.

>>> In potential infinity
>>> there are only definable numbers.
>>> But in actual infinity,
>>> there is more.
>>
>> If
>> you can get to p,
>> and
>> you can get to q,
>> then
>> you can get to s = p+q
>> and
>> you can get to k = (s-1)(s-2)/2+p
>>
>> If
>> you can get to k
>> then
>> you can get to s = ⌈(((8k+1)¹ᐟ²+1)/2)⌉
>> and
>> you can get to p = k-(s-1)(s-2)/2
>> and
>> you can get to q = s-p
>>
>> That is done without any use of _more_
>
> That is done without
> crossing the abyss of darkness.
> Not even ℵo/100000 is reached.

And all the fractions match all the indexes,
with no _more_

Your dark numbers do not enter the picture.

On 9/20/2022 1:35 PM, Chris M. Thomasson wrote:
> On 9/19/2022 7:13 PM, Sergi o wrote:
[...]

I got a direct solution, Sergio!

The direct solution does not need a successor!

i[n] = floor((1/3) * 10^(n+2) + 1)

Here is a snippet of a little program I wrote:
_____________________________________
// sci.math experiment
{ std::cout << "sci.math experiment\n" << std::endl;

// recursive solution
{
std::cout << "recursive solution" << std::endl;

unsigned long result = 34;
unsigned long n = 5;

for (unsigned long i = 0; i < n; ++i)
{
std::cout << "i[" << i << "] = " << result << "\n";

result = result + 3 * std::pow(10, i + 2);
}
}

// Direct solution
{
std::cout << "\ndirect solution" << std::endl;

unsigned long n = 5;

for (unsigned long i = 0; i < n; ++i)
{
unsigned long result = std::floor((1. / 3.) * std::pow(10,
i + 2) + 1);
std::cout << "i[" << i << "] = " << result << "\n";
}
}

}
_____________________________________

Here is the output:
_____________________________________
sci.math experiment

recursive solution
i[0] = 34
i[1] = 334
i[2] = 3334
i[3] = 33334
i[4] = 333334

direct solution
i[0] = 34
i[1] = 334
i[2] = 3334
i[3] = 33334
i[4] = 333334
_____________________________________

On 9/20/2022 4:25 AM, FromTheRafters wrote:
> It happens that Chris M. Thomasson formulated :
>> On 9/19/2022 3:51 PM, Sergi o wrote:
>>> On 9/19/2022 5:07 PM, Chris M. Thomasson wrote:
>>>> On 9/19/2022 12:01 PM, Jim Burns wrote:
>>>>> On 9/19/2022 8:51 AM, Timothy Golden wrote:
>>>>>> On Sunday, September 18, 2022
>>>>>> at 3:36:24 PM UTC-4, Jim Burns wrote:
>>>> [...]
>>>>> I don't know what the ellipsis in 333...34
>>>>> means. I've got a couple guesses, but they
>>>>> don't make much sense.
>>>>> And you are here to tell me what it means.
>>>>> So, why guess?
>>>>>
>>>>> What does 333...34 mean?
>>>> [...]
>>>>
>>>> Something like the following expansion.
>>>>
>>>> 34
>>>> 334
>>>> 3334
>>>> 33334
>>>> 333334
>>>> 3333334
>>>> 33333334
>>>> ...
>>>>
>>>> ;^)
>>>
>>> apply a geometric recursive formula;
>> [...]
>>
>> Okay, let me give it a go:
>> _______________________________
>> i[0] = 34
>> i[n] = i[n-1] + 3*10^(n+1)
>> _______________________________
>>
>> Therefore we get the following infinite expansion:
>> _______________________________
>> i[0] = 34
>> i[1] = i[0] + 3*10^(1+1) = 34 + 300 = 334
>> i[2] = i[1] + 3*10^(2+1) = 334 + 3000 = 3334
>> i[3] = i[2] + 3*10^(3+1) = 3334 + 30000 = 33334
>> ...
>> _______________________________
>>
>> that is 333...34, Right?
>
> This intended representation should also give the number of 'missing'
> threes (digits) otherwise it is too vague.

Fwiw, here is a direct solution:

i[n] = floor((1/3) * 10^(n+2) + 1)

On 9/19/2022 4:06 PM, Jim Burns wrote:
> On 9/19/2022 6:07 PM, Chris M. Thomasson wrote:
>> On 9/19/2022 12:01 PM, Jim Burns wrote:
[...]
> So, 333...34 isn't a natural number.
> What is it?

This is it:

i[n] = floor((1/3) * 10^(n+2) + 1)

Chris M. Thomasson wrote:

> On 9/19/2022 4:06 PM, Jim Burns wrote:
>> On 9/19/2022 6:07 PM, Chris M. Thomasson wrote:
>>> On 9/19/2022 12:01 PM, Jim Burns wrote:
> [...]
>> So, 333...34 isn't a natural number.
>> What is it?
>
> This is it: i[n] = floor((1/3) * 10^(n+2) + 1)

Heal and protect your prostate, give you deep, peaceful sleep.
https://www.bitchute.com/video/INJkZVuVFOol/

Chris M. Thomasson brought next idea :
> On 9/19/2022 4:06 PM, Jim Burns wrote:
>> On 9/19/2022 6:07 PM, Chris M. Thomasson wrote:
>>> On 9/19/2022 12:01 PM, Jim Burns wrote:
> [...]
>> So, 333...34 isn't a natural number.
>> What is it?
>
> This is it:
>
> i[n] = floor((1/3) * 10^(n+2) + 1)

Good, so you have it narrowed down to an infinite proper subset of the
natural numbers?

On 9/20/2022 3:23 PM, FromTheRafters wrote:
> Chris M. Thomasson brought next idea :
>> On 9/19/2022 4:06 PM, Jim Burns wrote:
>>> On 9/19/2022 6:07 PM, Chris M. Thomasson wrote:
>>>> On 9/19/2022 12:01 PM, Jim Burns wrote:
>> [...]
>>> So, 333...34 isn't a natural number.
>>> What is it?
>>
>> This is it:
>>
>> i[n] = floor((1/3) * 10^(n+2) + 1)
>
> Good, so you have it narrowed down to an infinite proper subset of the
> natural numbers?

I think so. I was thinking about an infinite a source of 3's... Well, a
good source of the threes is 1/3. Move them over, and floor it! Add one
to get the 4, and bam! We have it.