Sergio has brought this to us :
> On 9/18/2022 2:09 PM, FromTheRafters wrote:
>> Sergio wrote :
>>> On 9/18/2022 8:55 AM, Timothy Golden wrote:
>>>> On Saturday, September 17, 2022 at 6:53:57 PM UTC-4, Jim Burns wrote:
>>>>> On 9/17/2022 5:34 PM, Sergio wrote:
>>>>>> On 9/17/2022 1:57 PM, Jim Burns wrote:
>>>>>
>>>>>>> A process has last and first states,
>>>>>>> and _also_ each split of its states
>>>>>>> has last-before and first-after states.
>>>>>>> A FISON has last and first numbers,
>>>>>>> and each split of its numbers
>>>>>>> has last-before and first-after numbers.
>>>>>>>
>>>>>>> A FISON can represent a process.
>>>>>>
>>>>>> A FISON is a specific set of elements.
>>>>>>
>>>>>> a FISON can be used in a process,
>>>>>> but a FISON is not a process.
>>>>> I'm not disagreeing with you.
>>>>> A FISON is not a process.
>>>>>
>>>>> but neither is a number a herd of sheep.
>>>>> We can use a number to represent a herd of
>>>>> sheep, or at least some aspect of a herd of
>>>>> sheep.
>>>>>
>>>>> Also, I know that we use the word "process"
>>>>> to describe things which _don't_ have
>>>>> last-before and first-after for each split.
>>>>>
>>>>> In my limited understanding, _float glass_
>>>>> is produced by pouring molten onto some
>>>>> molten metal with a low melting-point, such
>>>>> as tin. I suspect that there are no steps
>>>>> such as adding a column of numbers has.
>>>>> https://en.wikipedia.org/wiki/Float_glass
>>>>>
>>>>> I don't have any other examples of float-glass
>>>>> style processes in mind right now, but I feel
>>>>> confident that there are many.
>>>>>
>>>>> However,
>>>>> a FISON can _represent_ certain processes,
>>>>> within my restricted sense of "process".
>>>>> I don't see anything to be concerned about.
>>>>> It seems to me that most of what mathematics
>>>>> is put to use for is _representing_
>>>>>
>>>>>> Sets do not change.
>>>>>
>>>>> Oh, I agree.
>>>>> Whether WM agrees gets tricky.
>>>>>
>>>>> WM has certainly said that sets do not change.
>>>>>
>>>>> What is "change" for WM?
>>>>> I'm exaggerating for effect:
>>>>> If,
>>>>> at one point, we say "It can fly"
>>>>> (true, because we refer to a plane)
>>>>> and
>>>>> at another point, we say "It can't fly"
>>>>> (true, because we refer to a boat),
>>>>> then
>>>>> whatever "it" refers to has _changed_
>>>>> from a plane to a boat.
>>>>>
>>>>> That's my currently-best guess at what
>>>>> WM is thinking. The usual caveats apply.
>>>>>
>>>>> Perhaps? WM sees a variable as needing
>>>>> some act of being pointed at a value within
>>>>> its domain. I think we all agree that can be
>>>>> done (in principle) for a FISON.
>>>>> I think WM claims that Cantor claims that
>>>>> it can be done for infinite collections.
>>>>>
>>>>>
>>>>> I offer an alternative to that notion of
>>>>> "variable": the notion of a reference to
>>>>> _one of_ a domain, finite or infinite,
>>>>> and of claims that we know are true without
>>>>> knowing _which_ one of the domain is
>>>>> referred to.
>>>>>
>>>>> We are finite. We can only express and
>>>>> manipulate finitely-many finite sentences.
>>>>> But referring is not expressing and not
>>>>> manipulating. Our finiteness does not
>>>>> keep us from referring to infinitely-many.
>>>>
>>>> In that induction is the process that allows these discussions then we
>>>> can induce these large values. For instance:
>>>>     333...34
>>>> is well defined.
>>>
>>> Not at all, it takes on an infinite range of values.
>>>
>>>   Please fix it and post back here for grading.
>>
>> I've seen notation similar to that, but his is definitely not a prime
>> example.
>>
>> https://en.wikipedia.org/wiki/Largest_known_prime_number#History_of_largest_known_prime_numbers
>
> when the number is specifically known, like M1279 sure, but his is not well
> defined.

and it is even.

WM <askasker48@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)

> 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 thought so, but then why did you correct your own words? Induction
proves properties true for all natural numbers.

--
Ben.

fredag 16 september 2022 kl. 09:32:01 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 15. September 2022 um 11:35:18 UTC+2:
> > onsdag 14 september 2022 kl. 20:41:56 UTC+2 skrev WM:
> > > zelos...@gmail.com schrieb am Mittwoch, 14. September 2022 um 19:05:29 UTC+2:
> > >
> > > > it works the way it works and just because you are too retarded to understand
> > > Show your superiority! Where are the O's when all X
> > > XOOO...
> > > XOOO...
> > > XOOO...
> > > XOOO...
> > > ...
> > > have been distributed according to Cantor's prescription k = (m + n - 1)(m + n - 2)/2 + m?
> > >
> > You are thinking in terms of a step by step process. That is WRONG
> Why?

Because WE ARE NOT COMPUTING IN MATHEMATICS

> >
> > The function f: N-> Q and such exist in its entirety already.
> Every definable natnumber can be reached step by step by its FISON.

No one CARES and your "definable" is IRRELEVANT

> > Your X and O is IRRELEVANT!
> The cry of a tortured creature.

It is a statement of fact, they are not relevant.

>
> Regards, WM

On Sunday, 18 September 2022 at 15:44:37 UTC-3, WM wrote:
[...]
> Every definable n can be reached by the process 1, 2, 3, ..., n.

*EVERY* natural number can be "reached" in this manner, even (Skewes' number)^^...^(Skewes' number) with (Skewes' number) '^'. That is how the set of natural numbers is *DEFINED*, you fucking imbecile.

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", Du psychotischer Spinner, but a "sequence" of numbers.

It seems that you just have discovered the fact that for any natural number n there are n - 1 smaller natural numbers (with IN = {1, 2, 3, ...}).

An INCREDIBLE fact!

On Sunday, September 18, 2022 at 3:36:24 PM UTC-4, Jim Burns wrote:
> On 9/18/2022 9:55 AM, Timothy Golden wrote:
> > On Saturday, September 17, 2022
> > at 6:53:57 PM UTC-4, Jim Burns wrote:
> >> On 9/17/2022 5:34 PM, Sergio wrote:
> >>> On 9/17/2022 1:57 PM, Jim Burns wrote:
>
> >>>> A process has last and first states,
> >>>> and _also_ each split of its states
> >>>> has last-before and first-after states.
> >>>> A FISON has last and first numbers,
> >>>> and each split of its numbers
> >>>> has last-before and first-after numbers.
> >>>>
> >>>> A FISON can represent a process.
> >>>
> >>> A FISON is a specific set of elements.
> >>>
> >>> a FISON can be used in a process,
> >>> but a FISON is not a process.
> >>> Sets do not change.
> >>
> >> Oh, I agree.
> >> Whether WM agrees gets tricky.
> >> I offer an alternative to that notion of
> >> "variable": the notion of a reference to
> >> _one of_ a domain, finite or infinite,
> >> and of claims that we know are true without
> >> knowing _which_ one of the domain is
> >> referred to.
> >>
> >> We are finite. We can only express and
> >> manipulate finitely-many finite sentences.
> >> But referring is not expressing and not
> >> manipulating. Our finiteness does not
> >> keep us from referring to infinitely-many.
> >
> > In that induction is the process that allows
> > these discussions then we can induce
> > these large values. For instance:
> > 333...34
> > is well defined. It's successor is:
> > 333...35
> > and we can even take the product of these two.
> > It is done via induction.
> What if we were to consider the infinite
> by carefully describing the finite and then
> saying "Not that"?
>
> That is the approach I'm taking.
>
> For each natural number n,
> an ordered set ⟨0,1,⋯,n-1,n⟩ exists
> with two ends 0 and n, first and last,
> and,
> for each _split_
> ( for each partition into two pieces,
> ( all of one before all of the other,
> there is a number i last-before the split,
> there is a number j first-after the split,
> and j = i+1
> > In that induction is the process that allows
> > these discussions then we can induce
> > these large values. For instance:
> > 333...34
> > is well defined.
> It's very unclear to me what '333...34'
> is intended to mean.
>
> If there is an ordered set ⟨0,⋯,333...34⟩
> in which, for each split,
> there is a pair i,i+1 across that split,
> then I would say, yes, that's what we mean
> by "natural number".
> If not, then not.
>
> ----
> The idea behind my definition is that,
> if
> ⟨0,⋯,333...34⟩ represents the procedure
> of counting from 0 to 333...34
> and
> if that procedure only ever advances
> from i to i+1, for some i,
> and
> if there is a split of ⟨0,⋯,333...34⟩
> then
> there is some way i,i+1 across that split.

Yes. I think this is exactly what we think of when we speak of infinity.
It feels as though we cannot get there, yet there are actually so many of them that once you name a few to try and name them all feels a bit silly.
That induction is to blame: this is why the value has such pricey appearances. That a part of your expression ⟨0,1,⋯,n-1,n⟩ exists in that ellipsis is entirely true. Until you can see that the ellipsis and induction are one and the same thing, well, I take pity on you.

>
> Otherwise,
> you can't get there (333...34) from here (0).
>
> ----
> Induction is a property of each end n
> of a set ⟨0,1,⋯,n-1,n⟩ with i,i+1 for splits.
>
> | Assume ⟨0,1,⋯,n-1,n⟩ exists.
> | Assume P(0) and ~P(n)
> |
> | There is a split of ⟨0,1,⋯,n-1,n⟩
> | such that
> | i is last-before the split,
> | i+1 is first-after the split,
> | and P(i) and ~P(i+1)
>
> Let's abbreviate
> "an ordered set ⟨0,1,⋯,n-1,n⟩ exists etc etc"
> as
> 'n ∈ ℕ'
>
> | If n ∈ ℕ exists, P(0) and ~P(n)
> | then i ∈ ℕ exists, P(i) and ~P(i+1)
>
> With some easy manipulation, that gives us:
>
> | If
> | P(0) and, for all i ∈ ℕ, P(i) -> P(i+1),
> | then,
> | for all n ∈ ℕ, P(n)
>
> That is the familiar statement of induction.
>
> ----
> For each ⟨0,1,⋯,n-1,n⟩
Well, when you inserted the ellipsis in here it is no different than when I inserted it into a digital value.
Anywhere they take a valid interpretation they are fair game, right? Is there some law of exclusion of ellipses?
Yeah, when we speak in finite terms it is probably true that the ellipsis should not come into use.

Speaking in terms of n and speaking in terms of n+1 will do just fine, thank you.
Somehow that fact that there is a bottom end of these things and that there is where the action is: you really are missing out.
I, Increment En, do hearby swear allegiance to the Ellipsis, without which we fail to succeed; without which life would take so little meaning.
That I, computationally am bound to certain rules that I cannot escape even if I try does expose my position in spacetime as limited and therefor trivial in its extent, and my successors, too. This sort of trivialization is broadly applicable. The entire galaxy is not going to develop wings and fly away. It is a thermodynamic problem. Yes, birds will evolve on planets with atmospheres and they can fly high into the sky. When the gamma ray burst happens we're gonna wish we were racoons living underground. I miss Going Underground. https://www.rt.com/shows/going-underground/

> n is in ⟨0,1,⋯,n-1,n⟩ but
> n+1 is not in ⟨0,1,⋯,n-1,n⟩
>
> Let ℕ be the set of
> all ends n of all ⟨0,1,⋯,n-1,n⟩
>
> If
> n ∈ ℕ
> then
> n ends ⟨0,1,⋯,n-1,n⟩
>
> If
> n ends ⟨0,1,⋯,n-1,n⟩
> then
> n+1 ends ⟨0,1,⋯,n-1,n⟩⊕⟨n+1⟩
> (⟨0,1,⋯,n-1,n⟩ with ⟨n+1⟩ appended)
> and
> ⟨0,1,⋯,n-1,n⟩⊕⟨n+1⟩ is also an ordered set
> with two ends, 0 first,
> and,
> for each _split_
> there is a number i last-before the split,
> there is a number i+1 first-after the split.
> so
> n+1 ∈ ℕ
>
> What if we were to consider the infinite
> by carefully describing the finite and then
> saying "Not that"?
>
> For each ⟨0,1,⋯,n-1,n⟩
> there is some n in ⟨0,1,⋯,n-1,n⟩
maybe use m here for the 'some n' since n is already in use.
> for which n+1 is not in ⟨0,1,⋯,n-1,n⟩
>
> Let's say that ⟨0,1,⋯,n-1,n⟩ is _finite_
>
> For ℕ
> there is NO n in ℕ
> for which n+1 is not in ℕ
>
> ℕ is not like any ⟨0,1,⋯,n-1,n⟩
> ℕ is not finite.

Well, in fact had you written ⟨0,1, 2, ⋯⟩
then you would have N, no?
In applications of N it seems that unity does take interpretation. Also it seems that in applications of N we need not worry about infinity at all. You can use as large a number as you need, and typically if you can't do your work with one of them then there is little hope of getting it done at all. In truth as we go to large n in physics we go gray since there is no perfect counter of say atoms in the universe. Approximations do just fine.

Still, it is possible to present a value such as
333...34
and perform computations on this value. In this regard I, and you, can touch high n, but it is done with regard for the inductive nature of the value.
That you care so much about large n and yet refuse to use any at all is a peculiar state of being. It's like you've gone comatose via the very thought of it. I seriously doubt if you have a problem just stopping at a million. Good luck even getting there with these old fashioned means. My computer is so fast I've had to go past a million to 100 million which it completes in two seconds. That you take some sense of superiority by teetering at the edge of N is understandable in these times. I did have an assignment in the code. Without that in 4 seconds my computer is counting to 1,000,000,000. I doubt that this is humanly possible without some sort of insanity kicking in. Please, don't try this at home. Somehow y'all feel that you've gone above and beyond with your language, yet where are the consequences?

I think the condition of inductive reasoning marks a stage of development of mathematics. I do trivialize it too much. It is a wonderful thing, especially in its applied form. I wonder to what degree is the sense of conflict around the new infinite constants a denial of working modulo ten?
At some level the naturalist understands that a presumed constant buried in all of his numbers is not natural at all. At such a stage I challenge you to present a value beyond nine, sir. At that junction then will digital analysis kick in for those who like to play in such largess? It's not considered fair game? Obviously not, since I have the first rendition of an instantiated infinite value. With no comment from the learned ones. Even when it is put directly under their nose.

Click here to read the complete article

On Monday, September 19, 2022 at 7:59:56 AM UTC-4, Gus Gassmann wrote:
> On Sunday, 18 September 2022 at 15:44:37 UTC-3, WM wrote:
> [...]
> > Every definable n can be reached by the process 1, 2, 3, ..., n.
> *EVERY* natural number can be "reached" in this manner, even (Skewes' number)^^...^(Skewes' number) with (Skewes' number) '^'. That is how the set of natural numbers is *DEFINED*, you fucking imbecile.

It is very appropriate how you put 'reached' in quotes. There is a halting problem. The natural numbers are inductively defined. Even a variable length computing program such as sage will eventually be brought to its knees when ram and flash or hard drive space are fully consumed. Any sense of need of a larger number actually does physically involve constructing a larger counter. Any claim of need of something larger and that it is not good enough is not substantiated by these means. Especially not by feeble humans. That the mathematician has written himself a ticket of purity and stamped it on his forehead means nothing. Start counting, sir. We'll watch your beard hairs grow and then they'll have you straight-jacketed and off to the asylum at about 123,456, which you will be laughing at hysterically. Meanwhile, I've set my computer to counting to 100,000,000,000 this morning. It's still working on it. Should be done at around 400 seconds as I recall, but I'm not even timing it.

Is it true that those who concern themselves with the natural numbers in this classical way do not care for computers?

On 9/19/2022 8:10 AM, Timothy Golden wrote:
> On Monday, September 19, 2022 at 7:59:56 AM UTC-4, Gus Gassmann wrote:
>> On Sunday, 18 September 2022 at 15:44:37 UTC-3, WM wrote:
>> [...]
>>> Every definable n can be reached by the process 1, 2, 3, ..., n.
>> *EVERY* natural number can be "reached" in this manner, even (Skewes' number)^^...^(Skewes' number) with (Skewes' number) '^'. That is how the set of natural numbers is *DEFINED*, you fucking imbecile.
>
> It is very appropriate how you put 'reached' in quotes. There is a halting problem. The natural numbers are inductively defined. Even a variable length computing program such as sage will eventually be brought to its knees when ram and flash or hard drive space are fully consumed. Any sense of need of a larger number actually does physically involve constructing a larger counter. Any claim of need of something larger and that it is not good enough is not substantiated by these means. Especially not by feeble humans. That the mathematician has written himself a ticket of purity and stamped it on his forehead means nothing. Start counting, sir. We'll watch your beard hairs grow and then they'll have you straight-jacketed and off to the asylum at about 123,456, which you will be laughing at hysterically. Meanwhile, I've set my computer to counting to 100,000,000,000 this morning. It's still working on it. Should be done at around 400 seconds as I recall, but I'm not even timing it.
>
> Is it true that those who concern themselves with the natural numbers in this classical way do not care for computers?
no.
Think of counting a very very very large number of sheeps. Like Africa large. have them go through a small place for counting, then into Italy. THey can
spread out from there. To save your memory space, just write the number of the sheep onto the sheep. Problem solved.

Ben Bacarisse schrieb am Montag, 19. September 2022 um 02:03:45 UTC+2:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
>
> > In potential infinity all definable natnumbers are all
> > natnumbers.
> I thought so, but then why did you correct your own words? Induction
> proves properties true for all natural numbers.

In general mathematics and in my book that is true. In Cantor's actual infinity it is wrong. Proof is easiest done when asking where the O's are lost when transforming the well-known matrix

XOO...
XOO...
XOO...
....

according to Cantor's prescription into

XXX...
XXX...
XXX...
....

But you are not wishing to talk about this embarrassing example?

Regards, 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?

> a "sequence" of numbers.

To go through a sequence means a process.
>
> 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.

Regards, WM

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:

>> The idea behind my definition is that,
>> if
>> ⟨0,⋯,333...34⟩ represents the procedure
>> of counting from 0 to 333...34
>> and
>> if that procedure only ever advances
>> from i to i+1, for some i,
>> and
>> if there is a split of ⟨0,⋯,333...34⟩
>> then
>> there is some way i,i+1 across that split.
>
> Yes.
> I think this is exactly what we think of
> when we speak of infinity.
>
> It feels as though we cannot get there,
> yet there are actually so many of them that
> once you name a few
> to try and name them all feels a bit silly.

There are two concepts in play here.
| You can't get there from here.
and
| There is no there.

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

This is not the same as claiming that
there are no other places, except in
the narrow sense of there being
no other places _which we are reasoning about_

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

What I mean by that is
there is a step i,i+1 from one side
to the other of that spit.

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

In order to get
from 0, where P(0) is true
to n, where P(n) is false,
you must be able to get there from here

If there is no step i,i+1 for which
P(i) is true and
P(i+1) is false,
then you can't get to any n for which
P(n) is false.

> That induction is to blame: this is why
> the value has such pricey appearances.
> That a part of your expression ⟨0,1,⋯,n-1,n⟩
> exists in that ellipsis is entirely true.
> Until you can see that the ellipsis and
> induction are one and the same thing,
> well, I take pity on you.

Induction and ellipsis are different.
Ellipsis is a way of _not_ expressing some
sequence, like 0,1,2,3,4...
or like 1,1,2,3,5,8,13,21,34,...

Ellipsis '...' implicitly claims that
the sequence is too obvious to need stating.
'...' == "You know what I mean here"

That's perfectly fair when used correctly,
when something really is too obvious to
need stating. It's helpful to clear away
as much of the haystack as you can for your
reader, to offer them the best chance of
finding the needle of your purpose
hidden in the verbiage.

Ellipsis can be used incorrectly.
Mostly when something's NOT too obvious
to need stating. But that _should_ be easily
fixed. Whoever is using ellipsis thinks
something is obvious. What does the ellipsis
stand for, please?

>> For each ⟨0,1,⋯,n-1,n⟩
>
> Well, when you inserted the ellipsis in here
> it is no different than when I inserted it
> into a digital value.

I'm using the ellipsis there as part of
the name of a set.

I hope that it's obvious what it means there,
but I can tell you what I mean.
In fact, I've already told you:
for each split, some i,i+1 crosses it.

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?

> Yeah, when we speak in finite terms
> it is probably true that
> the ellipsis should not come into use.

No, the ellipsis means the same in a
finite sequence: "You know what I mean here".
In different places, "what I mean" is
different.

On 9/19/2022 12:11 PM, WM wrote:
> 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.

Wrong, they are sequences, not processes. Fail

>
> 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?

No, that quote does not support your claim, fail.

>
>> a "sequence" of numbers.
>
> To go through a sequence means a process.

Wrong.

By your rules you call any arrangement of numbers, or letters, "a process". Such is your loose and undisciplined thinking.

>>
>> 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.

you call bad math a discovery, and you have 3 errors in your last sentence.

>
> Regards, WM

On 9/19/2022 12:07 PM, WM wrote:
> Ben Bacarisse schrieb am Montag, 19. September 2022 um 02:03:45 UTC+2:
>> WM <askas...@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>>
>>> In potential infinity all definable natnumbers are all
>>> natnumbers.
>> I thought so, but then why did you correct your own words? Induction
>> proves properties true for all natural numbers.
>
> In general mathematics and in my book that is true. In Cantor's actual infinity it is wrong.

If Cantor is wrong, why do you run to his sayings as the ultimate truth all the time ?

You only include Cantors name to make your silly posts more important. Stop it.

Proof is easiest done when asking where the O's are lost when transforming the well-known matrix
>
> XOO...
> XOO...
> XOO...
> ...
>
> according to Cantor's prescription into
>
> XXX...
> XXX...
> XXX...
> ...
>
> But you are not wishing to talk about this embarrassing example?
>
> Regards, WM

It is embarrassing for you to post such obvious nonsense.

Remember, what you post does not go away, and is world wide.

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
....

;^)

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;

so if a(1) = 34

then a(2) = a(1) * 9.82352941176...

a(3) = a(2) * 9.9820359281437...

a(4) = a(3) * 9.998200359928014397...

a(5) = a(4) * 9.9998200035999280014399712

a(6) = a(5) * 9.99998200003599992800014399712

note the coefficients are expanding in an unusual way, adding 9, and 0 and 9 and 0 but keeping others....

try this;

334 - 34 = 300 so

a(1) = 34
a(2) = a(1) + 300
a(3) = a(2) + 3,000
a(4) = a(3) + 30,000

a(n) = a(n-1) + 3 * 10^n

not much else there.......

On 9/19/2022 6:07 PM, Chris M. Thomasson wrote:
> On 9/19/2022 12:01 PM, 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
> ...
>
> ;^)

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.

That's not the same as saying
333...34 does not exist,
but
_induction_ follows from
being able to get there from here.

So, 333...34 isn't a natural number.
What is it?

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?

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.

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:
>
>>> 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
>> ...
>>
>> ;^)
>
> 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.
>
> That's not the same as saying
> 333...34 does not exist,
> but
> _induction_ follows from
> being able to get there from here.
>
> So, 333...34 isn't a natural number.
> What is it?
>

It is a number that can be constructed, digit-by-digit:
_______________________________
i[0] = 34
i[n] = i[n-1] + 3*10^(n+1)
_______________________________

That expands to:
_______________________________
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
....
_______________________________

Not sure exactly what to call it, but there it is.

On 9/19/2022 6:06 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:
>
>>> 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
>> ...
>>
>> ;^)
>
> 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.
>
> That's not the same as saying
> 333...34 does not exist,
> but
> _induction_ follows from
> being able to get there from here.
>
> So, 333...34 isn't a natural number.
> What is it?
>

it is a natural number (no decimals)

.... means continue, so it cant be 333999934 or anything like that,

no info is known about the ..., except the 3's on each end

SO, 3's it is. just dont know how many of them...

even if your map says there is no way you can get there.

So, no matter where you go, there you are.

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....

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.

måndag 19 september 2022 kl. 19:07:36 UTC+2 skrev WM:
> Ben Bacarisse schrieb am Montag, 19. September 2022 um 02:03:45 UTC+2:
> > WM <askas...@gmail.com> writes:
> > (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> > Unendlichen" at Hochschule Augsburg.)
> >
> > > In potential infinity all definable natnumbers are all
> > > natnumbers.
> > I thought so, but then why did you correct your own words? Induction
> > proves properties true for all natural numbers.
> In general mathematics and in my book that is true. In Cantor's actual infinity it is wrong. Proof is easiest done when asking where the O's are lost when transforming the well-known matrix
>
> XOO...
> XOO...
> XOO...
> ...
>
> according to Cantor's prescription into
>
> XXX...
> XXX...
> XXX...
> ...
>
> But you are not wishing to talk about this embarrassing example?
>
> Regards, WM
Inducion is true and works in cantors as well because it makes no DIFFERENCE there!

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?
>
> > a "sequence" of numbers.
>
> To go through a sequence means a process.

No it doesn't because mathematics DO NOT DO PROCESSES!

> >
> > 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!

>
> Regards, WM

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:
>
> >> 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
> > ...
> >
> > ;^)
> 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.
Constructive freedom exposes our ability to develop this value.
Neither can you count to infinity.
This is consistent with a branching structure. Over and over I suggest that these concepts lay in a region sequestered from a first tier.
Yet whether induction itself is off in this place as well: I think this may be the can that is getting opened.
Call this curricular activity if you must, but those who wish to dabble in infinity are in the same boat as this value is.
The machine that I am on now counts to 18446744073709551616, and for some reason gnu is making it a terrible time getting the printf() format descriptor %lu. Obviously I took the wrong way through the internet. To what degree is information accumulation a problem? Not just on the internet but in mathematics as well? I am invoking digital analysis onto natural valued analysis. The two go in different styles really.

Were you able to count to n? So you can't work in n? I don't accept your criterion. In effect you have capped the natural numbers too early. You can construct a split atop your finite version, but then, by your own thinking isn't 333...34 then a dark number or some such? I know WM rejects it because it does have a successor, but the idea of some number out beyond the naturals seems to be what this is by your own thinking extended here by myself.

The split values you mention are used by me to do the computations. I enter into sage a value long enough to yield repetition. From there the problem is inductive. You have carefully left out the invocation of induction. To me the ellipsis implies induction and we are speaking of inductive values as a result. The rapid rate of the digital form versus the base one sequential form that usually emits from mathematicians is in some ways much more powerful. By the time we make the hundredth digit we are up into 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 of your usual sequence in n. The digits are well defined aren't they? As to who can't get there first; well clearly the digital analysis wins this race. It is far faster than the usual natural analysis. Therefor in games of chasing infinities the digital analysis wins.

These values will sum and multiply. Digital infinity exposes the qualities of infinity better than your natural analysis does. Does infinity have a largest digit? A smallest digit? Are there two sides to every natural value? And best of all how many infinities are there? That we can securely expose an array of infinities is quite a nice feature. That they even have order is amazing. This will entail the usage of an aleph mark that is foreign to existing mathematics ultimately, but I do believe that it can be said that:
111...11 < 222...22
It can be said that the RHS is twice the LHS. Yet both are so large as to be uncountable, shall we say? Even the product of these two is possible:
246913580...197530864...2
and so a class of infinity of doubly induced values is born. Possibly there is a digit there in the middle that I am missing. I'd have a bit more work to verify this, but the inductive nature of this computation is exposed here:

sage: a = 111111111111111111111111111111111111111111111111111111111111111111111
.....: 1111111111111111111111111111111111111111111111111111111111111111111111111
.....: 1111111111111111111111111111111111111111111111111111111111111111111111111
.....: 1111111111111111111111111111111111111111111111111111111111111111111111111
.....: 11111111111111111111111111111111111111111111111111111111111111111111111
sage: b = 222222222222222222222222222222222222222222222222222222222222222222222
.....: 2222222222222222222222222222222222222222222222222222222222222222222222222
.....: 2222222222222222222222222222222222222222222222222222222222222222222222222
.....: 2222222222222222222222222222222222222222222222222222222222222222222222222
.....: 2222222222222222222222222222222222222222222222222222222222222222222222222
.....: 22
sage: a*b
2469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135802469135777775308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308641975308642
sage:

>
> That's not the same as saying
> 333...34 does not exist,
> but
> _induction_ follows from
> being able to get there from here.

And I am indeed getting there from here by induction. You cannot refuse the inductive nature of these values. In terms of a halting problem all inductive usage can be challenged as a failure. When induction works best cancellations occur and the result works well. Here it is more of a child-like thing. Sure enough it is the same thing over and over. Possibly I am misusing the term, but as well I think existing mathematics is misusing it too. The ambiguity is dripping from these threads which clog sci.math.

>
> So, 333...34 isn't a natural number.
> What is it?

Wow. Thanks Jim for weighing in. That's great. Do you agree that it has a successor?
That it is discrete in nature? It's awfully close to the traditional version.
And here you are wrapping the same iteration problem up the other way.
As if to say that your ray gun has this mode but mine does not.

Induction ultimately states that if it works in n, and it works in n+1, then it is universally true. The language 'it works' is vague, but then, so is induction. In this case we are literally computing with inductive values. There will be no surprises. We feel secure in this reasoning.

I don't actually like it myself, except that it does address some of the nagging infinity discussions, doesn't it? I believe that what is in dispute is induction itself and how many times and ways it gets invoked. Ellipses for me imply induction. Iteration implies induction when done by a human on paper rather than by a machine. And no: machines do not always halt. I hit control-c on my last counting attempt because I didn't want to wear out my processor. The poor thing, caught in such a tight loop, must have been getting awfully dizzy.