Mathematics, science and Abraham Robinson

[David Petry]

Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).

"I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)

That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.

Even serious mathematicians often tell me that they have no interest in that idea, and since mathematics is by definition what mathematicians are interested in, the idea has no role to play in mathematics.

Here's another quote from Mr. Robinson.

"Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless." (A. Robinson)

Exactly right. Someday, mathematicians will wake up.

Here's a quote from Yuri Manin, another outstanding mathematician.

"Georg Cantor's grand meta-narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory." (Y. Manin)

I mention that because I've been asked why I usually mention Cantor by name when I say that set theory doesn't really belong in mathematics. I think that he (Cantor) is ultimately responsible.

The mathematics that is undeniably of value to the world is the mathematics that can be understood as a science.

[zelos...@gmail.com]

set theory is part of matematics and very important in it

[FromTheRafters]

David Petry has brought this to us :

> Here's a quote from Abraham Robinson, who is undeniably a legit mathematician
> (use Wikipedia if you need to know more).
>
> "I think that there is a real need, in formalism and elsewhere, to link our
> understanding of mathematics with our understanding of the physical world."
> (A. Robinson)
>
> That's exactly what I have been saying for years, and the response I get from
> guys like Malum, Messager, and Burns is that the idea is stupid, crackpot,
> and motivated by evil intentions. There are nothing but crackpots in this
> newsgroup.
>
> Even serious mathematicians often tell me that they have no interest in that
> idea, and since mathematics is by definition what mathematicians are
> interested in, the idea has no role to play in mathematics.

Many mathematicians aren't interested in philosophy.

[zelos...@gmail.com]

torsdag 12 maj 2022 kl. 04:37:18 UTC+2 skrev david...@gmail.com:

> Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).
>
> "I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)
>
> That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.

There are plenty of us sane ones, I am and a few others. You, Gabriel, etc are the cranks and crackpots here.

>
> Even serious mathematicians often tell me that they have no interest in that idea, and since mathematics is by definition what mathematicians are interested in, the idea has no role to play in mathematics.

It doesn't because mathematicians do not care about the real world and physical stuff.

>
> Here's another quote from Mr. Robinson.
>
> "Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless." (A. Robinson)
>
> Exactly right. Someday, mathematicians will wake up.

There is nothing to wake up from. Infinities work in mathematics and mathematicians know it is not a physical thing and guess what? We do not care.

>
> Here's a quote from Yuri Manin, another outstanding mathematician.
>
> "Georg Cantor's grand meta-narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory." (Y. Manin)
>
> I mention that because I've been asked why I usually mention Cantor by name when I say that set theory doesn't really belong in mathematics. I think that he (Cantor) is ultimately responsible.
>
> The mathematics that is undeniably of value to the world is the mathematics that can be understood as a science.

It has value because of what it is, it lose value if it goes down the path you are proposing.

[sobriquet]

On Thursday, May 12, 2022 at 4:37:18 AM UTC+2, david...@gmail.com wrote:
> Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).
>
> "I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)
>
> That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.
>
> Even serious mathematicians often tell me that they have no interest in that idea, and since mathematics is by definition what mathematicians are interested in, the idea has no role to play in mathematics.
>
> Here's another quote from Mr. Robinson.
>
> "Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless." (A. Robinson)

In a way they do exist. Like if you add up an infinite number of numbers, the result might be
a number (like the sum of numbers obtained from 1/(2^n) for n=0 to n=infinity, which
adds up to 2).
Even though you can't actually obtain the result by summing up an infinite number of
numbers, there are alternative ways to obtain the result that you would get if you could
add up an infinite number of numbers.

Though perhaps this can be refuted with an argument from physics where
we might be able to obtain evidence that any region of space or spacetime can't
be infinitely subdivided (analogous to the way you can't keep subdividing
a quantity of gold or other forms of energy or matter into infinitesimally small
quantities).

[FromTheRafters]

sobriquet laid this down on his screen :

> On Thursday, May 12, 2022 at 4:37:18 AM UTC+2, david...@gmail.com wrote:
>> Here's a quote from Abraham Robinson, who is undeniably a legit
>> mathematician (use Wikipedia if you need to know more).
>>
>> "I think that there is a real need, in formalism and elsewhere, to link our
>> understanding of mathematics with our understanding of the physical world."
>> (A. Robinson)
>>
>> That's exactly what I have been saying for years, and the response I get
>> from guys like Malum, Messager, and Burns is that the idea is stupid,
>> crackpot, and motivated by evil intentions. There are nothing but crackpots
>> in this newsgroup.
>>
>> Even serious mathematicians often tell me that they have no interest in that
>> idea, and since mathematics is by definition what mathematicians are
>> interested in, the idea has no role to play in mathematics.
>>
>> Here's another quote from Mr. Robinson.
>>
>> "Infinite totalities do not exist in any sense of the word (i.e., either
>> really or ideally). More precisely, any mention, or purported mention, of
>> infinite totalities is, literally, meaningless." (A. Robinson)
>
> In a way they do exist. Like if you add up an infinite number of numbers, the
> result might be a number (like the sum of numbers obtained from 1/(2^n) for
> n=0 to n=infinity, which adds up to 2).
> Even though you can't actually obtain the result by summing up an infinite
> number of numbers, there are alternative ways to obtain the result that you
> would get if you could add up an infinite number of numbers.

That's what many people overlook. Also, I'm hard-pressed to find a real
number which doesn't have at least one convergent series
representation.

[FredJeffries]

Gentlemen, gentlemen. PLEASE.

Even though you can't do something we can still obtain the result that we would get if we could do it?!

What kind of silliness is that? How do we KNOW what we would get if we can't do it? Not only can't do it, but have no idea what it would mean to do it?

We cannot 'add up an infinite number of numbers'. It's not (merely) that there are too many of them. It's that there is no last term in the series. The process of repeated binary addition never stops. Talk about an END of the process or what happens when the process FINISHES is just nonsense.

Finding the limit of the sequence of partial sums of an infinite series is NOT an 'alternative ways to obtain the result that you would get if you could add up an infinite number of numbers'.

It is an entirely DIFFERENT species of task/process altogether.

The 'sum of an infinite series', i.e. the limit of the sequence of partial sums, is the (one, unique) number which best characterizes that series as a whole. It is a 'universal' number for that series. It is the number which best SUMmarizes the series.

Finding the limit of the sequence of partial sums is what we do INSTEAD of adding up all of the terms because the notion of 'adding up all of the terms' is at best currently undefined and at worst complete nonsense. And all talk of 'adding up all of the terms' only gives the cranks and trolls more grist for their mills.


https://en.wikipedia.org/wiki/Universal_property

[David Petry]

On Thursday, May 12, 2022 at 4:43:47 AM UTC-7, zelos...@gmail.com wrote:
> torsdag 12 maj 2022 kl. 04:37:18 UTC+2 skrev david...@gmail.com:
> > Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).


> > "I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)


> > That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.

> There are plenty of us sane ones, I am and a few others. You, Gabriel, etc are the cranks and crackpots here.

Recall what I've said many times: the serious purpose of mathematics is to provide a conceptual framework that helps us reason about the real world.

So, (question for Zelos), how is that different from what Robinson said? And if you admit that there is no difference, then do you think that Robinson was a crank and a crackpot?

[sergio]

and....

[sergio]

On 5/16/2022 9:50 PM, David Petry wrote:
> On Thursday, May 12, 2022 at 4:43:47 AM UTC-7, zelos...@gmail.com wrote:
>> torsdag 12 maj 2022 kl. 04:37:18 UTC+2 skrev david...@gmail.com:
>>> Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).
>
>
>>> "I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)
>
>
>>> That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.
>
>> There are plenty of us sane ones, I am and a few others. You, Gabriel, etc are the cranks and crackpots here.
>
>
> Recall what I've said many times: the serious purpose of mathematics is to provide a conceptual framework that helps us reason about the real world.

corrected;

"A serious purpose of mathematics is to provide a conceptual framework that helps us model the real world".

>
> So, (question for Zelos), how is that different from what Robinson said? And if you admit that there is no difference, then do you think that Robinson was a crank and a crackpot?

this one is BS;

[zelos...@gmail.com]

tisdag 17 maj 2022 kl. 04:50:54 UTC+2 skrev david...@gmail.com:
> On Thursday, May 12, 2022 at 4:43:47 AM UTC-7, zelos...@gmail.com wrote:
> > torsdag 12 maj 2022 kl. 04:37:18 UTC+2 skrev david...@gmail.com:
> > > Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).
>
>
> > > "I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)
>
>
> > > That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.
>
> > There are plenty of us sane ones, I am and a few others. You, Gabriel, etc are the cranks and crackpots here.
> Recall what I've said many times: the serious purpose of mathematics is to provide a conceptual framework that helps us reason about the real world.

No one cares what you, a crank, says.

>
> So, (question for Zelos), how is that different from what Robinson said? And if you admit that there is no difference, then do you think that Robinson was a crank and a crackpot?

He might have been a crank. I know for certain that you are. But both of you are wrong here.

[FromTheRafters]

sergio has brought this to us :

I fail to see how this refutes what I said. We have the real numbers
because we want to use the ideas of 'convergence' and 'limit' to define
our numbers. Summation is usually done one way for finite summations
and another way for infinite summations but both are summations.

[sergi o]

seems like FredJeffries has never read Newton, nor had a class on convergence, or he forgot it all.
I think he misquoted Robison or did not understand it..

[sobriquet]

We know it by mathematical induction . So given an area of, say 2, we know we can subdivide one of its constituent areas into two equal areas of 1 (the base case). Then we apply the assumption that we can further subdivide one of its two smallest constituent areas into two smaller equal areas 1/2 and 1/2 and we continue this step recursively indefinitely.
Either we end up with a number that can't be further subdivided into smaller numbers and we found the smallest possible number, or there is nothing which prevents us from further subdividing one of the two smallest numbers into two equal smaller numbers.

https://en.wikipedia.org/wiki/Mathematical_induction

I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.

Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.

[WM]

sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:

> I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
>
> Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.

Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.

All numbers you get by induction have ℵo successors before ω:
∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
They cannot be exhausted, because they remain always there.

But the set of all natural numbers exhausts also these successors:
|ℕ \ {1, 2, 3, ...}| = 0
or
{0, 1, 2, 3, ..., ω} \ ℕ = {0, ω}.

Regards, WM

[FredJeffries]

Inasmuch as no one understood the point I was trying to make, I will only apologize for being such a bad expositor and wasting your time with my semi-rant. I will not pursue the matter further.

On to bookkeeping points: I am not responsible for the Robinson (mis)quote in this thread. That should be obvious not only from the indentations and headings, but because I would have included the source and context of the statement. Most importantly, I would not have omitted the remainder of Robinson's statement (see below).

The statement is from Abraham Robinson's (one of the most important mathematicians of the 20th century, by the way) talk at the 1964 International Congress for Logic, Methodology, and Philosophy of Science in Jerusalem titled simply 'Formalism 64'.

Alas, neither the transcription of that talk nor the proceedings of the congress seem to be electronically available.

A few years later, Robinson published 'From a formalist's point of view' in 'Dialectica' Vol. 23, No. 1 (1969), pp. 45-49 where he restated and clarified points he had made in that talk. This article is available for those with JSTOR access
https://www.jstor.org/stable/42968450

For a secondary account we may consult Robinson's biographer Joseph Dauben (yes, the same Dauben who wrote the acclaimed biography of Georg Cantor). There is an available article 'Abraham Robinson and Nonstandard Analysis: History, Philosophy, and Foundations of Mathematics'

https://conservancy.umn.edu/bitstream/handle/11299/185659/11_07Dauben.pdf

In the section 'Robinson and "Formalism 64"' (p. 10), Dauben discusses the quote in question.

<quote>
Robinson emphasized two factors in rejecting his earlier Platonism in favor of a formalist position:

(i) Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless.

(ii) Nevertheless, we should continue the business of Mathematics 'as usual,' i.e. we should act as if infinite totalities really existed.
</quote>

[sobriquet]

On Wednesday, May 18, 2022 at 2:42:22 PM UTC+2, WM wrote:
> sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:
>
> > I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
> >
> > Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.
> Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.
>
> All numbers you get by induction have ℵo successors before ω:
> ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
> They cannot be exhausted, because they remain always there.

But then what does it mean when we prove something with mathematical induction?
I would assume that if we proof a property P(n) for natural numbers n with induction, that
means that for all natural numbers n, P(n) holds.
But since there is an infinite set of natural numbers, the property has been proven for an actual
infinite number of cases.

So if we prove that we can subdivide a given finite area into an arbitrary natural number of
parts decreasing in size, and the parts sum up to the original area we started
out with, that means we know that the actual infinite sum of subdivided areas decreasing
in size indefinitely yields the original area.

[WM]

sobriquet schrieb am Mittwoch, 18. Mai 2022 um 18:25:04 UTC+2:
> On Wednesday, May 18, 2022 at 2:42:22 PM UTC+2, WM wrote:
> > sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:
> >
> > > I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
> > >
> > > Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.
> > Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.
> >
> > All numbers you get by induction have ℵo successors before ω:
> > ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
> > They cannot be exhausted, because they remain always there.

> But then what does it mean when we prove something with mathematical induction?

We prove it for all definable natural numbers.

> I would assume that if we proof a property P(n) for natural numbers n with induction, that
> means that for all natural numbers n, P(n) holds.

You know that the set of numbers which this is proved for is always finite?

> But since there is an infinite set of natural numbers, the property has been proven for an actual
> infinite number of cases.

By induction we can prove that the numbers reached by induction have ℵo successors before ω.Therefore they belong to a finite set. It is impossible to subdivide ℕ into two consecutive infinite aleph_0-sets.The successor numbers cannot be identified by induction. They can only be used collectively.

The set of all natural numbers contains also these successors:
ℕ \ {1, 2, 3, ...} = { }

>
> So if we prove that we can subdivide a given finite area into an arbitrary natural number of
> parts decreasing in size, and the parts sum up to the original area we started
> out with, that means we know that the actual infinite sum of subdivided areas decreasing
> in size indefinitely yields the original area.

Yes that holds for every definable subdivision into n parts.

Regards, WM

[WM]

FredJeffries schrieb am Mittwoch, 18. Mai 2022 um 18:14:16 UTC+2:
>
> <quote>
> Robinson emphasized two factors in rejecting his earlier Platonism in favor of a formalist position:
>
> (i) Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless.
>
> (ii) Nevertheless, we should continue the business of Mathematics 'as usual,' i.e. we should act as if infinite totalities really existed.
> </quote>

Let's continue with the origianl source:

"I feel quite unable to grasp the idea of an actual infinite totality. To me there appears to exist an unbridgeable gulf between sets or structures of one, or two, or five elements, on one hand, and infinite structures on the other hand [...] I must regard a theory which refers to an infinite totality as meaningless in the sense that its terms and sentences cannot posses the direct interpretation in an actual structure that we should expect them to have by analogy with concrete (e.g., empirical) situations." [A. Robinson: "Formalism 64", North-Holland, Amsterdam, p. 230f]

Regards, WM

[FromTheRafters]

WM was thinking very hard :

Go ahead, we'll wait. While we are waiting, why don't you tell us what
'successors before ω' means here.

[Fritz Feldhase]

On Wednesday, May 18, 2022 at 8:32:16 PM UTC+2, WM wrote:

> Let's continue with the origianl source:
>
> "I feel quite unable to grasp the idea of an actual infinite totality. To me there appears to exist an unbridgeable gulf between sets or structures of one, or two, or five elements, on one hand, and infinite structures on the other hand [...] I must regard a theory which refers to an infinite totality as meaningless in the sense that its terms and sentences cannot posses the direct interpretation in an actual structure that we should expect them to have by analogy with concrete (e.g., empirical) situations." [A. Robinson: "Formalism 64", North-Holland, Amsterdam, p. 230f]

So what?! Do you psychotic asshole full of shit understand the significance of the statement

"(ii) Nevertheless, we should continue the business of Mathematics 'as usual,' i.e. we should act as if infinite totalities really existed." [A. Robinson]

?

Obviously not. After all, your "business" is crankery and psychotic mumbo-jumbo.

[Timothy Golden]

I wonder to what degree those concerned with natural valued infinity are possibly entertaining the birth of the continuum?
As you start plopping down positions on a line and they keep going at some large value the discernment of the smaller values seems imperceptible. Particularly thinking in terms of large n a relative position abstraction with arbitrarily fine granularity ensues.

This bears out as we consider the decimal value as a natural value. For instance as 1/3= 0.333... then dropping the decimal point we are dealing in a natural value 333.... Even a value such as 2/5 in perfection will yield 0.4000... which again as a natural value is 4000...
These sorts of infinite precision values are in denial of epsilon/delta theory, whereas 2/5=0.4000 is a finite precision instance. In other words this is a computationally valid instance and if we did wish to work in greater precision we could. Dropping the decimal from this instance we see 2/5 as 4000 and of course its data can be recovered by building up a structure xe where x is the natural value and e is the decimal place. The 'e' portion is arguably natural valued, but it is of a different meaning. It's position is a matter of placing a secondary unity upon the otherwise purely natural value.

In this way the continuum can be built by having a high regard for the natural value; augmenting it with a new sense of unity; and as well keeping a regard for epsilon/delta theory and its adjustable precision. In effect those presuming perfection in their rational values have been working with an infinite form of the natural value and never saw it as ambiguous. Could these be the infinite concerns of the natural valued philosophy? I do credit in part WM's dark number with helping me develop the gray number that is the continuum; the more truly 'real' value. That the interpretation could come back then onto the naturals in this way is an interesting turn. So then are values such as 333... dark values? Hah! I have managed to instantiate them!

[Fritz Feldhase]

On Wednesday, May 18, 2022 at 8:25:52 PM UTC+2, WM wrote:
> sobriquet schrieb am Mittwoch, 18. Mai 2022 um 18:25:04 UTC+2:
> >
> > But then what does it mean when we prove something with mathematical induction?
> >

> We prove it for all [...] natural numbers.

Or all elements in in IN.

Right.

> > I would assume that if we proof a property P(n) for natural numbers n with induction, that
> > means that for all natural numbers n, P(n) holds.

Using symbols: An(n e IN -> P(n)) or An e IN: P(n). Right!

> You know that the set of numbers which this is proved for is always finite?

No, we don't know that IN is finite.

Hint: "Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, ... ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), ... ." (Wikipedia)

> > But since there is an infinite set of natural numbers, the property has been proven for an actual infinite number of cases.

Yes. See the quote from Wikipedia.

[Quote from the German page: "Die vollständige Induktion ist eine mathematische Beweismethode, nach der eine Aussage für alle natürlichen Zahlen bewiesen wird, die größer oder gleich einem bestimmten Startwert sind. Da es sich um unendlich viele Zahlen handelt..."]

> [...] the numbers [that] have ℵo successors before ω [...] belong to a finite set.

Sure, each and every "number" n belongs to the finite set {n}. An incredible insight, Mückenheim!

On the other hand, each and every number in IN has ℵo successors "before ω" (< ω), but IN is infinite.

[sergi o]

On 5/18/2022 1:25 PM, WM wrote:
> sobriquet schrieb am Mittwoch, 18. Mai 2022 um 18:25:04 UTC+2:
>> On Wednesday, May 18, 2022 at 2:42:22 PM UTC+2, WM wrote:
>>> sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:
>>>
>>>> I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
>>>>
>>>> Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.
>>> Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.
>>>
>>> All numbers you get by induction have ℵo successors before ω:
>>> ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
>>> They cannot be exhausted, because they remain always there.
>
>> But then what does it mean when we prove something with mathematical induction?
>
> We prove it for all definable natural numbers.

wrong, it is provee for all natural numbers.

>
>> I would assume that if we proof a property P(n) for natural numbers n with induction, that
>> means that for all natural numbers n, P(n) holds.
>
> You know that the set of numbers which this is proved for is always finite?

wrong, he said all natural numbers. Did you miss that ?

>
>> But since there is an infinite set of natural numbers, the property has been proven for an actual
>> infinite number of cases.
>
> By induction we can prove that the numbers reached by induction have ℵo successors before ω.

wrong, By induction we can prove that the numbers reached by induction upto ω.

>Therefore they belong to a finite set.

Wrong, it is all natural numbers, the entire set, ℕ

> It is impossible to subdivide ℕ into two consecutive infinite aleph_0-sets.

wrong, odd and even... etc

> The successor numbers cannot be identified by induction.

You stopped at k again. Nobody stopped induction, it goes all the way out, like Chuck Norris.

>They can only be used collectively.

yes, used numbers after collectively harvested can be plowed into the fields like fertilizer.

>
> The set of all natural numbers contains also these successors:
> ℕ \ {1, 2, 3, ...} = { }

intentional diversion, and misdirection.

>>
>> So if we prove that we can subdivide a given finite area into an arbitrary natural number of
>> parts decreasing in size, and the parts sum up to the original area we started
>> out with, that means we know that the actual infinite sum of subdivided areas decreasing
>> in size indefinitely yields the original area.
>
> Yes that holds for every definable subdivision into n parts.

your "definable" is meaningless, with its beeps, flashes, raps, hoofs, giggles

>
> Regards, WM

[Fritz Feldhase]

On Wednesday, May 18, 2022 at 9:07:55 PM UTC+2, sergi o wrote:
> On 5/18/2022 1:25 PM, WM wrote:
> >

> > It is impossible to subdivide ℕ into two consecutive infinite [...] sets.

That's indeed true! (Big surprise!)

> wrong, odd and even...

Not "consecutive" (though infiite).

What he's talking about here is a partition of IN into two _infinite_ sets {a_1, a_2, a_3 ...} and {b_1, b_2, b_3, ...} such that

a_i < b_j, for all i.j e IN.

(/Partition/ of IN into two sets {a_1, a_2, a_3 ...}, {b_1, b_2, b_3, ...} here means: {a_1, a_2, a_3 ...} =/= {}, {b_1, b_2, b_3, ...} =/= {}, {a_1, a_2, a_3 ...} n {b_1, b_2, b_3, ...} = {} and {a_1, a_2, a_3 ...} u {b_1, b_2, b_3, ...} = IN.)

Need to see a proof? :-P

> > ... that holds for every definable subdivision into n parts.

> >
> your "definable" is meaningless, with its beeps, flashes, raps, hoofs, giggles

Indeed. :-)

[Timothy Golden]

On Wednesday, May 18, 2022 at 3:28:27 PM UTC-4, Fritz Feldhase wrote:
> On Wednesday, May 18, 2022 at 9:07:55 PM UTC+2, sergi o wrote:
> > On 5/18/2022 1:25 PM, WM wrote:
> > >
> > > It is impossible to subdivide ℕ into two consecutive infinite [...] sets.
>
> That's indeed true! (Big surprise!)

Well, going off of my new interpretation I can split them at:
333...33 versus 333...34.
How's that?

[Timothy Golden]

On Wednesday, May 18, 2022 at 3:58:31 PM UTC-4, Timothy Golden wrote:
> On Wednesday, May 18, 2022 at 3:28:27 PM UTC-4, Fritz Feldhase wrote:
> > On Wednesday, May 18, 2022 at 9:07:55 PM UTC+2, sergi o wrote:
> > > On 5/18/2022 1:25 PM, WM wrote:
> > > >
> > > > It is impossible to subdivide ℕ into two consecutive infinite [...] sets.
> >
> > That's indeed true! (Big surprise!)
> Well, going off of my new interpretation I can split them at:
> 333...33 versus 333...34.
> How's that?

More carefully:
..., 333...29, 333...30, 333...31, 333...32, 333...33 |split| 333...34, 333...35, 333...36, ...

[Fritz Feldhase]

On Wednesday, May 18, 2022 at 10:15:00 PM UTC+2, timba...@gmail.com wrote:
> On Wednesday, May 18, 2022 at 3:58:31 PM UTC-4, Timothy Golden wrote:
> > On Wednesday, May 18, 2022 at 3:28:27 PM UTC-4, Fritz Feldhase wrote:
> > > On Wednesday, May 18, 2022 at 9:07:55 PM UTC+2, sergi o wrote:
> > > > On 5/18/2022 1:25 PM, WM wrote:
> > > > >
> > > > > It is impossible to subdivide ℕ into two consecutive infinite [...] sets.
> > > > >
> > > That's indeed true! (Big surprise!)
> >
> > Well, going off of my new interpretation I can split them at:
> >

> ..., 333...29, 333...30, 333...31, 333...32, 333...33 |split| 333...34, 333...35, 333...36, ...

You are sure that you are talking about _natural_ numbers (i.e. the elements in IN) AND the usual order defined on them?

Say, n < m :<-> Ek e IN \ {0}: n + k = m ,

where + is the usual addition defined on IN.

[WM]

Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:56:22 UTC+2:

> Hint: "Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, ... ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), ... ." (Wikipedia)

This statement is wrong if there is omega existing. Between every number subject to induction and omega there are aleph_0 natural numbers which are not subject to induction because they do never vanish.

numbers [that] have ℵo successors before ω [...] belong to a finite set.
>
> Sure, each and every "number" n belongs to the finite set {n}.

If each and every number were incapable of spanning an actually infinite set, there could not exist an actually infinite set. But most natural numbers belong to a subset larger than every FISON.

> On the other hand, each and every number in IN has ℵo successors "before ω" (< ω), but IN is infinite.

Try to find your mistake. Hint: There are not two consecutive aleph_0-sets in IN.

Regards, WM

[WM]

Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:39:58 UTC+2:
> After all, your "business" is crankery and psychotic mumbo-jumbo.

You claim that every natural number has aleph_0 successors between itself and omega.
That implies that no natural number has less successors between itself and omega.
That implies that between all elements of the the set of all natural numbers and omega there are aleph_0 successors.
That implies that between the set of all natural numbers and omega there are aleph_0 successors.

The latter can only be denied by crankery and psychotic mumbo-jumbo. That is your business.

Regards, WM

[FromTheRafters]

Timothy Golden laid this down on his screen :

> On Wednesday, May 18, 2022 at 3:28:27 PM UTC-4, Fritz Feldhase wrote:
>> On Wednesday, May 18, 2022 at 9:07:55 PM UTC+2, sergi o wrote:
>>> On 5/18/2022 1:25 PM, WM wrote:
>>>>
>>>> It is impossible to subdivide ℕ into two consecutive infinite [...] sets.
>>
>> That's indeed true! (Big surprise!)
>
> Well, going off of my new interpretation I can split them at:
> 333...33 versus 333...34.
> How's that?

Not even close. :) They are partitioning sets and those aren't even
sets.

[FromTheRafters]

WM wrote :

> Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:39:58 UTC+2:
>> After all, your "business" is crankery and psychotic mumbo-jumbo.
>
> You claim that every natural number has aleph_0 successors between itself and
> omega.

You claim this. This means that you are accepting the successor
function (each and every element of N has one successor) and induction,
so don't try to disclaim it later.

[Timothy Golden]

To be honest with you I suspect that this form will eventually be treated as invalid. This form is already in use though it is not appreciated.
I am referring to the decimal value as when we write:
1/3 = 0.333...
and we observe that the decimal point of this representation may be treated as an augmentation to the natural value. It seems that to make it stick to these infinite forms we should explain its position from the left (high) side of the natural value. This would be to maintain consistency with these existing forms. Please note that the digits themselves haven't a care how small or large they may be. They are each connected to their adjacents in the same way regardless of their size. In this way, and as it is written above the form has been in use for some time already. All that is needed is to appreciate that the decimal point is an addition to the structure of the natural value. It is certainly easily removed. It's replacement is readily coded. In this case we'll specify the form
x e
where x is the natural value and e is the digital position of the decimal point with zero being at the far left of the value. This is of course to be deleted from the format now and we are left to deal with the rational value in natural terms. No dirty reradixers need apply themselves here. Everything is modulo 10, sir. Perfect values...

Of course the farce is with you and ends when you realize that the decimal format is in fact a gray format; in finite terms, where these ellipses will never require usage, we can play out epsilon/delta theory on any value merely by chasing digits. At least this is how I see it. As we pull the taffy the other way now it does seem to be working out in WM's favor. I guess we have a bifurcation, but I insist that it comes from real analysis. If you allow the ellipsis form 0.333... then you have chosen one side of the bifurcation. If you reject the ellipsis then the continuum goes gray. It's dark numbers, or it's gray numbers.

Oddly the notion of recovering the continuum from the naturals is here as well. As odd, under my interpretation the continuum always takes natural values too... augmented with a secondary unity aka the decimal point.

Could it be of interest that some quite complicated values like
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140799...

whose square is 2000... are available to work with... if you believe in infinite fairly tales. Remarkably though this is where that epsilon/delta business crept in in the first place. Why it would apply to some values but not to others is problematic.

[Timothy Golden]

Sorry I didn't answer your question directly: yes: addition works:
333...29 + 5 = 333...34
333...29 + 333...34 = 666...63

[Fritz Feldhase]

On Wednesday, May 18, 2022 at 11:36:34 PM UTC+2, timba...@gmail.com wrote:

> Sorry I didn't answer your question directly: yes: addition works:

> 333...29 + 5 = 333...34
> 333...29 + 333...34 = 666...63

You know there's a smallest natural number, usually called 0 (or 1).

Which of your symbols donotes 0 (or 1)?

And if we are already at that, if X denotes the natural number n which symbol denotes the natural number n+1?

You see there's a system of names which is called "unary"?

In this system 1 is denoted by the symbol "|" and if n is denoted by the symbol X, then n+1 is denoted by the symbol X appended by the symbol "|".

[Timothy Golden]

I don't get your nonsense. Let n = 333...34. then n+1 is 333...35. zero is still zero and one is still one. Already at what? we're discussing WM's dark numbers, roughly. As you dodge the usage of the ellipsis and play dumb, the rational value 1/3 = 0.333... is a similar instance. I've explained my analysis carefully enough. You talk past it as if it hasn't occurred; as if what you see is nonsense. It's for you to falsify, sir. Just talk straight, please.

[sergi o]

On 5/18/2022 3:55 PM, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:39:58 UTC+2:
>> After all, your "business" is crankery and psychotic mumbo-jumbo.
>
> You claim that every natural number has aleph_0 successors between itself and omega.

ok

> That implies that no natural number has less successors between itself and omega.

so you state 17 has as many successors as 100^100^100 ?

> That implies that between all elements of the the set of all natural numbers and omega there are aleph_0 successors.

Wrong. try re-wording that.

> That implies that between the set of all natural numbers and omega there are aleph_0 successors.

Wrong. The set is not a natural number, and neither is omega.

>
> The latter can only be denied by crankery and psychotic mumbo-jumbo. That is your business.

you are the one with the wrong implied mumbo-jumbo, but you are crank, and that is your business here.

>
> Regards, WM
>

[sergi o]

On 5/18/2022 3:48 PM, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:56:22 UTC+2:
>
>> Hint: "Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, ... ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), ... ." (Wikipedia)
>
> This statement is wrong if there is omega existing.

wrong, omega is not needed.

> Between every number subject to induction and omega there are aleph_0 natural numbers which are not subject to induction because they do never vanish.

"vanish" is not a mathematical term. Omega is diversion.

>
> numbers [that] have ℵo successors before ω [...] belong to a finite set.

wrong, no set is specified.

>>
>> Sure, each and every "number" n belongs to the finite set {n}.
>
> If each and every number were incapable of spanning an actually infinite set,

So, how does the element 17 span an infinite set ?

> there could not exist an actually infinite set.

hokie-pokie

> But most natural numbers belong to a subset larger than every FISON.

baloney, prove it. you are spouting nonsense.
use equations

>
>> On the other hand, each and every number in IN has ℵo successors "before ω" (< ω), but IN is infinite.
>
> Try to find your mistake. Hint: There are not two consecutive aleph_0-sets in IN.

another red herring.

>
> Regards, WM

[Fritz Feldhase]

On Thursday, May 19, 2022 at 12:54:22 AM UTC+2, timba...@gmail.com wrote:

> the rational value 1/3 = 0.333... is a similar instance.

Fuck off, you silly troll.

Hint: We were talking about NATURAL NUMBERS! GOT THAT, you silly asshole full of shit?!

*PLONK*

[Jim Burns]

On 5/18/2022 4:48 PM, WM wrote:
> Fritz Feldhase schrieb
> am Mittwoch, 18. Mai 2022 um 20:56:22 UTC+2:

>> Hint:
>> "Mathematical induction is a mathematical proof technique.
>> It is essentially used to prove that a statement P(n)
>> holds for every natural number n = 0, 1, 2, 3, ... ;
>> that is, the overall statement is a sequence of
>> infinitely many cases P(0), P(1), P(2), P(3), ... ."
>> (Wikipedia)
>
> This statement is wrong if there is omega existing.

No.

> Between every number subject to induction and omega
> there are aleph_0 natural numbers which are not
> subject to induction because they do never vanish.

You are facing in the wrong direction.
Stand on a number k finitely-many '+1" from 0.
Turn away from omega and look at 0.

k is subject to induction *because*
k is finitely-many '+1' from 0.

If the truth-value of P(i) changes, in going from P(0)
to ~P(k), then, somewhere, the truth-value must change
in a single step, in going from P(j) to ~P(j+1).
[1]

If the predicate P(i) does not permit the truth-value
to change in a single step, then the truth-value
cannot change in finitely-many steps, either.

This is natural-number, finitely-many-+1 induction.
As I've stated it here, induction says
| If k exists such that P(0) and ~P(k),
| then j exists such that P(j) and ~P(j+1)

That is equivalent to the usual formulation
| If P(0) and, for all j, P(j) -> P(j+1),
| then, for all k, P(k)

Infinity enters the story because there are
infinitely-many numbers which are finitely-many '+1'
from 0.

[1]
If the truth-value of P(i) changes, in going from P(0)
to ~P(k), then, somewhere, the truth-value must change
in a single step, in going from P(j) to ~P(j+1).

Assume that P(0) and ~P(k), and
k is finitely-many '+1" from 0.

Define _finitely-many_ to mean that,
for each split BEFORE and AFTER of ⟨0,...,k⟩
some j ends BEFORE and j+1 begins AFTER.

Define AFTERnot to be the collection of all i in ⟨0,...,k⟩
such that ~P(i) or i > i' such that ~P(i').

Define BEFOREnot to be the collection of all i in ⟨0,...,k⟩
such that i is not in AFTERnot.

P(0), so BEFOREnot is not empty.
~P(k), so AFTERnot is not empty.

BEFOREnot ∪ AFTERnot = ⟨0,...,k⟩
Each of BEFOREnot is before each of AFTERnot.

Therefore, BEFOREnot and AFTERnot is a split of ⟨0,...,k⟩

_Because k is finite_
some j ends BEFOREnot and j+1 begins AFTERnot.

Given the definitions of BEFOREnot and AFTERnot,

any i in ⟨0,...,k⟩ for which ~P(i) is in AFTERnot
j is in BEFOREnot
P(j)

j+1 is first in AFTERnot.
Either ~P(j+1)
or, for some i' in BEFOREnot, ~P(i')
However,
there is no i' in BEFOREnot, ~P(i')
~P(j+1)

Therefore,
P(j) and ~P(j+1)

[Fritz Feldhase]

On Wednesday, May 18, 2022 at 10:48:58 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:56:22 UTC+2:
> >
> > Hint: "Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, ..."
> >

> wrong [...]. Between every number subject to induction and omega there are [...] natural numbers which are not subject to induction

Fascinating, you psychotic asshole full of shit.

So in your psychotic kingdom

P(0) & An(n e IN & P(n) -> P(n+1))

does NOT imply

An e IN: P(n) ?

Hint: An e IN: P(n) means that THERE IS NO natural number n such that P(n) does NOT HOLD.

Fuck off, you silly crank!

[Fritz Feldhase]

On Thursday, May 19, 2022 at 2:30:59 AM UTC+2, Jim Burns wrote:

> You are facing in the wrong direction.

Concerning Mückenheim that's _always_ the case.

[sergi o]

?

[zelos...@gmail.com]

onsdag 18 maj 2022 kl. 14:42:22 UTC+2 skrev WM:
> sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:
>
> > I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
> >
> > Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.
> Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.
>
> All numbers you get by induction have ℵo successors before ω:
> ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
> They cannot be exhausted, because they remain always there.
>

> But the set of all natural numbers exhausts also these successors:
> |ℕ \ {1, 2, 3, ...}| = 0
> or
> {0, 1, 2, 3, ..., ω} \ ℕ = {0, ω}.
>
> Regards, WM
this is meaningless for what you want to do

[zelos...@gmail.com]

onsdag 18 maj 2022 kl. 20:25:52 UTC+2 skrev WM:
> sobriquet schrieb am Mittwoch, 18. Mai 2022 um 18:25:04 UTC+2:
> > On Wednesday, May 18, 2022 at 2:42:22 PM UTC+2, WM wrote:

> > > sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:
> > >
> > > > I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
> > > >
> > > > Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.
> > > Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.
> > >
> > > All numbers you get by induction have ℵo successors before ω:
> > > ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
> > > They cannot be exhausted, because they remain always there.
>

> > But then what does it mean when we prove something with mathematical induction?
> We prove it for all definable natural numbers.

your "definable" is MEANINGLESS

> > I would assume that if we proof a property P(n) for natural numbers n with induction, that
> > means that for all natural numbers n, P(n) holds.
> You know that the set of numbers which this is proved for is always finite?

> > But since there is an infinite set of natural numbers, the property has been proven for an actual
> > infinite number of cases.

> By induction we can prove that the numbers reached by induction have ℵo successors before ω.Therefore they belong to a finite set. It is impossible to subdivide ℕ into two consecutive infinite aleph_0-sets.The successor numbers cannot be identified by induction. They can only be used collectively.

False, they belong to N which is not finite.

>
> The set of all natural numbers contains also these successors:
> ℕ \ {1, 2, 3, ...} = { }

> >
> > So if we prove that we can subdivide a given finite area into an arbitrary natural number of
> > parts decreasing in size, and the parts sum up to the original area we started
> > out with, that means we know that the actual infinite sum of subdivided areas decreasing
> > in size indefinitely yields the original area.

> Yes that holds for every definable subdivision into n parts.
>
> Regards, WM

[Timothy Golden]

Wow, that brings passive aggressive to a whole new level!

The values under discussion here are infinite forms right?

Is 333...3 an infinite form? Yes. Are ellipses in use in mathematics? Yes.
Are they used in number theory? Yes.
It happens that the series of digits does not really care what size they are. They act based on their adjacency.
That a decimal point might or might not be added to the structure: it is just this fact that it is an addition to an existing structure and that it has its own distinct character that allows me to provide this working interpretation.
It is readily stated that
333...33 + 1 = 333...34
so I've satisfied your requirement. Now you've completely blown off the argument why exactly?
It seems you can't handle it.
It's pretty clear though what else collapses as you formalize your rejection rather than simply send emotional rejection my way.
Such an emotional man must surely be sensitive enough to this argumentation.
Your rejection is well received, but as to the technical falsification: this is where I care.

[WM]

Jim Burns schrieb am Donnerstag, 19. Mai 2022 um 02:30:59 UTC+2:
> On 5/18/2022 4:48 PM, WM wrote:

> > Between every number subject to induction and omega
> > there are aleph_0 natural numbers which are not
> > subject to induction because they do never vanish.
> You are facing in the wrong direction.

So it appears to someone who never faced in the right direction. What is wrong with my direction? (Except that it disproves your beliefs.)

> Stand on a number k finitely-many '+1" from 0.
> Turn away from omega and look at 0.
>
> k is subject to induction *because*
> k is finitely-many '+1' from 0.

That does not exclude aleph_0 dark numbers immediately before omega.

>
> Infinity enters the story because there are
> infinitely-many numbers which are finitely-many '+1'
> from 0.

Impossible if actual infinity is meant, that is *more* than any finite number of +1.

Every finite number of +1 is contained in a potentially infinite set which is followed by an infinite set - if infinity is actual.

Regards, WM

[Timothy Golden]

On Wednesday, May 18, 2022 at 6:17:15 PM UTC-4, Fritz Feldhase wrote:
> On Wednesday, May 18, 2022 at 11:36:34 PM UTC+2, timba...@gmail.com wrote:
>
> > Sorry I didn't answer your question directly: yes: addition works:
>
> > 333...29 + 5 = 333...34
> > 333...29 + 333...34 = 666...63
> You know there's a smallest natural number, usually called 0 (or 1).
>
> Which of your symbols donotes 0 (or 1)?

I haven't done anything to the notation. I have no idea where this question comes from.

>
> And if we are already at that, if X denotes the natural number n which symbol denotes the natural number n+1?
>
> You see there's a system of names which is called "unary"?
>
> In this system 1 is denoted by the symbol "|" and if n is denoted by the symbol X, then n+1 is denoted by the symbol X appended by the symbol "|".

I really don't see where you are going with this. I guess you are saying this is the use of IN in the prior text?

A long time ago I did ask for an instance of a dark number from WM. But W was was busy mucking with something.
Now through a long devolution I see it. But my own theory actually rejects the use of the ellipses and lands us on a continuum that always obeys epsilon/delta theory. This is upheld by physical correspondence. That this theory of the continuum now comes back and interacts at the natural value is of interest I believe, for it is as well a part of my theory that we fundamentally represent the continuum with the natural value. It is just that it has to be augmented with a new interpretation of unity.

Simply, and from a fairly bland starting point: observe that the rational value 3/5 is actually two values and an operator. In this way we have just falsified the rational value as fundamental in half a sentence. Furthermore that operator (division) is actually an inverse operator and so is even less fundamental than it already wasn't. Furthermore the notion of closure is nonexistent so long as those two values are treated as naturals. In effect we see a heap of lies that pass as the birthing of the continuum for the status quo.

Now we extend a bit further, realizing that another form of representation exists which does not embed any operator: the decimal value. So we go ahead and actually compute the value 3/5 and we get 0.6! What we see is a new format of numerical representation. Most view this form as if the dot acts as a separator of digits, as if the connection of the digits is somehow broken there, but it is not at all broken there. The chain of digits has not a care as to where that little dot goes. Instead what we see is a mark of unity; a secondary unity that does not even need to challenge the first form of unity which is that of the natural number, and that is all that is left when we remove the little dot known to most in this day as the decimal point.

It's a can of worms that takes many tendrils of interpretation, but the fact that they can even hold up into here; into WM's theory is really quite a strange event imo. The issues of precision that put up a direct wall between the engineers and physicists on the one side and the mathematicians on the other has to be a false wall. By this reasoning the ellipsis form will not be accepted. Accepting a bifurcation here though, and by this allowing in the well accepted
1/3 = 0.333...
it is not much of a stretch to admit that
1/3 = 0.333...33
is it? This is interesting in that numbers are somewhat two-sided entities. But really the magic is done when we pull the decimal point. This is the nature of structured thought which the computer programmer understands well. He who obeys a strict compiler knows that such things must be done. And of course as to what compiler the mathematicians have been obeying... well, it does not look so good and strict through this lens does it?
No. Mathematics is a soft version of compilation and ultimately every check has been done by habituated humans who are passing along their rules under threat of failure; not just making monkeys of yourselves but guaranteeing that your inner monkey is locked down to boot. This is how programmable humans are, and the lack of attention to this detail within mathematics is a horrible abuse, as if to say that philosophy needn't have any part of this cult. Nor need the physicist nor the engineer usher his complaint about the perfection presumed of the rational value upon his continuum... until the hardware computer arises. Then too, jotting down the computations on a piece of paper we see the same thing: the numbers are all natural valued. They may contain a little dot somewhere, and you'd better keep track of it, but that is all.

So we can defend WM's theory now by instantiating constant valued infinite instances. This is something he himself has failed to do.
All of you who are caught in n and never care to instantiate a damn thing are fraught with a sense of superiority that fails you.
This is a big win in the short term here for me. Yet in the long term I do believe that it is the collapse of much of existing mathematical theory.
Whether your sort can think this big; well, no, I suppose not. You all have proven it time and time again. The practice of applying simple logic and falsifying somebody elses application of simple logic simply has not occurred. Slur away, please.

[FromTheRafters]

Timothy Golden laid this down on his screen :

> On Wednesday, May 18, 2022 at 8:22:28 PM UTC-4, Fritz Feldhase wrote:
>> On Thursday, May 19, 2022 at 12:54:22 AM UTC+2, timba...@gmail.com wrote:
>>
>>> the rational value 1/3 = 0.333... is a similar instance.
>> Fuck off, you silly troll.
>>
>> Hint: We were talking about NATURAL NUMBERS! GOT THAT, you silly asshole
>> full of shit?!
>>
>> *PLONK*
>
> Wow, that brings passive aggressive to a whole new level!
>
> The values under discussion here are infinite forms right?
>
> Is 333...3 an infinite form? Yes. Are ellipses in use in mathematics? Yes.
> Are they used in number theory? Yes.

Something similar is used to denote large primes, and it is not an
infinite form used this way.

[sergi o]

he called you a silly troll, which I disagree with.

But I have to ask, what type of program are you using to generate such large volumes of non-applicable fluff ?

[sergi o]

On 5/19/2022 8:44 AM, WM wrote:
> Jim Burns schrieb am Donnerstag, 19. Mai 2022 um 02:30:59 UTC+2:
>> On 5/18/2022 4:48 PM, WM wrote:
>
>>> Between every number subject to induction and omega
>>> there are aleph_0 natural numbers which are not
>>> subject to induction because they do never vanish.
>> You are facing in the wrong direction.
>
> So it appears to someone who never faced in the right direction. What is wrong with my direction?

your direction always is 'not math'.

>
>> Stand on a number k finitely-many '+1" from 0.
>> Turn away from omega and look at 0.
>>
>> k is subject to induction *because*
>> k is finitely-many '+1' from 0.
>
> That does not exclude aleph_0 dark numbers immediately before omega.

there are no dark numbers.

>>
>> Infinity enters the story because there are
>> infinitely-many numbers which are finitely-many '+1'
>> from 0.
>
> Impossible if actual infinity is meant, that is *more* than any finite number of +1.

sorry, infinity is bigger than finite.

>
> Every finite number of +1 is contained in a potentially infinite set which is followed by an infinite set - if infinity is actual.

meaningless drivel.

>
> Regards, WM
>

[Jim Burns]

On 5/19/2022 9:44 AM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 19. Mai 2022 um 02:30:59 UTC+2:
>> On 5/18/2022 4:48 PM, WM wrote:

>>> Between every number subject to induction and omega
>>> there are aleph_0 natural numbers which are not
>>> subject to induction because they do never vanish.
>>
>> You are facing in the wrong direction.
>
> So it appears to someone who never faced in the
> right direction. What is wrong with my direction?
> (Except that it disproves your beliefs.)
>
>> Stand on a number k finitely-many '+1" from 0.
>> Turn away from omega and look at 0.
>>
>> k is subject to induction *because*
>> k is finitely-many '+1' from 0.
>
> That does not exclude aleph_0 dark numbers
> immediately before omega.

Both dark numbers and omega are irrelevant to
natural-number induction.
It applies where it applies because,
for each BEFORE and AFTER =< where it applies,

some j ends BEFORE and j+1 begins AFTER.

What excludes dark numbers immediately before omega
is the definition of omega as the first infinite
ordinal.

>> Infinity enters the story because there are
>> infinitely-many numbers which are finitely-many '+1'
>> from 0.
>
> Impossible if actual infinity is meant,

You (WM) do not use "actual infinity" in the usual way.

How you use it is self-contradictory, so, yes,
meaning it that way produces impossibilities.
Of course, that's not what you think you're saying.

> that is *more* than any finite number of +1.

Each finite number is less than
as many as all finite numbers there are.

[sergi o]

On 5/18/2022 3:14 PM, Timothy Golden wrote:
> On Wednesday, May 18, 2022 at 3:58:31 PM UTC-4, Timothy Golden wrote:
>> On Wednesday, May 18, 2022 at 3:28:27 PM UTC-4, Fritz Feldhase wrote:
>>> On Wednesday, May 18, 2022 at 9:07:55 PM UTC+2, sergi o wrote:
>>>> On 5/18/2022 1:25 PM, WM wrote:
>>>>>
>>>>> It is impossible to subdivide ℕ into two consecutive infinite [...] sets.
>>>
>>> That's indeed true! (Big surprise!)
>> Well, going off of my new interpretation I can split them at:

>> 333...33 versus 333...34.
>> How's that?
>

> More carefully:

> ..., 333...29, 333...30, 333...31, 333...32, 333...33 |split| 333...34, 333...35, 333...36, ...

whoa! you should have used |chop|

>
>>>
>>>> wrong, odd and even...
>>>
>>> Not "consecutive" (though infiite).
>>>
>>> What he's talking about here is a partition of IN into two _infinite_ sets {a_1, a_2, a_3 ...} and {b_1, b_2, b_3, ...} such that
>>>
>>> a_i < b_j, for all i.j e IN.
>>>
>>> (/Partition/ of IN into two sets {a_1, a_2, a_3 ...}, {b_1, b_2, b_3, ...} here means: {a_1, a_2, a_3 ...} =/= {}, {b_1, b_2, b_3, ...} =/= {}, {a_1, a_2, a_3 ...} n {b_1, b_2, b_3, ...} = {} and {a_1, a_2, a_3 ...} u {b_1, b_2, b_3, ...} = IN.)
>>>
>>> Need to see a proof? :-P
>>>
>>>>> ... that holds for every definable subdivision into n parts.
>>>>>
>>>> your "definable" is meaningless, with its beeps, flashes, raps, hoofs, giggles
>>> Indeed. :-)

[WM]

Jim Burns schrieb am Donnerstag, 19. Mai 2022 um 19:23:45 UTC+2:
> On 5/19/2022 9:44 AM, WM wrote:

> > That does not exclude aleph_0 dark numbers
> > immediately before omega.
> Both dark numbers and omega are irrelevant to
> natural-number induction.

Of course. Induction covers only the collection of definable natural numbers.

That does not exclude aleph_0 dark numbers immediately before omega.

> You (WM) do not use "actual infinity" in the usual way.

I use it in the correct way: A quantity larger than all finite quantities. "In spite of significant difference between the notions of the potential and actual infinite, where the former is a variable finite magnitude, growing above all limits, the latter a constant quantity fixed in itself but beyond all finite magnitudes, it happens deplorably often that the one is confused with the other." [Cantor, p. 374]

But you succeed always to distract from the clear evidence proving dark fractions:

We check the number of indexes by bijecting them with the fractions of the first column (we could use every other column or line as well). When applying the indexes for indexing fractions such that m/n gets the index
k = (m + n - 1)(m + n - 2)/2 + m
with the resulting sequence
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
then the integer fractions have to supply these indexes. They are stripped off these indexes. The number of not indexed fractions remains constant.

All definable fractions get indexed. Most fractions don 't get indexed. Dark fractions.

Regards, WM

[WM]

timba...@gmail.com schrieb am Donnerstag, 19. Mai 2022 um 16:10:04 UTC+2:

> A long time ago I did ask for an instance of a dark number from WM.

Those are number whihc can on,y be defined collectively.

For every definable number we find ℵo successors
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
after n and before omega.

For all natural numbers, defined collectively, we find no successors
ℕ \ {1, 2, 3, ...} = { } .

This proves the existence numbers which can only be defined collectively.

Regards, WM

[sergi o]

On 5/19/2022 1:32 PM, WM wrote:
> Jim Burns schrieb am Donnerstag, 19. Mai 2022 um 19:23:45 UTC+2:
>> On 5/19/2022 9:44 AM, WM wrote:
>
>>> That does not exclude aleph_0 dark numbers
>>> immediately before omega.
>> Both dark numbers and omega are irrelevant to
>> natural-number induction.
>

<snip crap>

>
>> You (WM) do not use "actual infinity" in the usual way.
>

<snip crap>
>
> Regards, WM

[sergi o]

On 5/19/2022 2:10 PM, WM wrote:
> timba...@gmail.com schrieb am Donnerstag, 19. Mai 2022 um 16:10:04 UTC+2:
>
>> A long time ago I did ask for an instance of a dark number from WM.
>

<snip crap>
>
> Regards, WM
>
>

[Jim Burns]

On 5/19/2022 2:32 PM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 19. Mai 2022 um 19:23:45 UTC+2:
>> On 5/19/2022 9:44 AM, WM wrote:

>>> That does not exclude aleph_0 dark numbers
>>> immediately before omega.
>>
>> Both dark numbers and omega are irrelevant to
>> natural-number induction.
>
> Of course.
> Induction covers only the collection of
> definable natural numbers.

Natural numbers are only definable ==
only finitely-many '+1' from 0 ==
only k such that, for each BEFORE and AFTER =< k,

some j ends BEFORE and j+1 begins AFTER.

This is why induction does not have any exceptions
where it is intended to be applied, in the naturals
== the definable naturals.

> That does not exclude aleph_0 dark numbers
> immediately before omega.

Something else excludes dark numbers immediately
before omega, the definition of omega.

>> You (WM) do not use "actual infinity" in the usual way.
>
> I use it in the correct way:

A WM-potentially-infinite collection can match
a proper subset. It lacks Bob-conservation.

A WM-actually-infinite collection _cannot_ match
a proper subset, but one of its subsets is
WM-potentially-infinite.

There are no WM-actually-infinite collections.

> I use it in the correct way:
> A quantity larger than all finite quantities.
> "In spite of significant difference between the notions
> of the potential and actual infinite, where the former
> is a variable finite magnitude, growing above all limits,
> the latter a constant quantity fixed in itself but
> beyond all finite magnitudes, it happens deplorably often
> that the one is confused with the other."
> [Cantor, p. 374]

A claim about _one of_ a collection which is true
no matter which one of the collection is referred to
can be used to _beat the bounds_ of the collection.

We know that how this method proceeds from one claim
to the next stays inside the bounds of the collection.

We can describe an infinite collection by describing
_one of_ the intended collection and then saying
that's true of each of them.

We can reason about an infinite collection by
reasoning from our description of _one of_ them.

The difference between potential and actual is
a pointless distinction as far as this method is
concerned. We _begin_ complete with respect to
whatever we're talking about, and we _stay_ complete
until we've finished (for now) reasoning.

> But you succeed always to distract from the clear
> evidence proving dark fractions:

...

> All definable fractions get indexed.
> Most fractions don 't get indexed.
> Dark fractions.

All definable fractions get indexed ==
all fractions get indexed.
No dark fractions.

[Timothy Golden]

'collectively' is a terminology that seems uncontroversial and acceptable. To what degree we work collectively here on usenet I wouldn't care to say, yet somehow it does hold true. Clearly definitions only work within their collective. If I reject natural numbers then I am outside of the collective that works with natural numbers and so your condition is somewhat a tautological situation.

You've chosen to delete the relevant input of mine here. I suggest that I have instantiated a dark number and you choose to ignore it. This is not a collective situation. Once again I will put the data back in:
333...33, 333...34, 333...35
are three adjacent infinite values aren't they? Is it a collective of three? I suppose. There are issues. Add two of them and land at 666...67 but add all three and land at 1000...02 then add in the first again to get 4333...35? Some of the addition seems to work uncontroversially, but this last bit seems like trouble. Anyway the ability to present a digital representation of your dark numbers does not mean that we should be able to do too much with them. It seems pretty clear that 666...6 is twice as large as 333...3 doesn't it?

And of course these values gain their legitimacy from the old well accepted 1/3=0.333...3.
Well, close enough...
is good enough?

The question of legitimacy of these values I do want to discuss. That these happen to come very close to some of the conditions of dark numbers does seem interesting. Regardless of their darkness they are infinite values. Their legitimacy imo was built when the mathematician accepted 0.3333... as a valid representation. Falsify one and you falsify the other. I actually do accept this falsification but on your side of things it seems as though you'd want to let these large values in. The only difference between the decimal value and the natural value is a little dot in the notation. That little dot is a piece of data that has been augmented to the natural value. Of course I already said all this. It's not so carefully stated here.

[David Petry]

On Wednesday, May 18, 2022 at 9:14:16 AM UTC-7, FredJeffries wrote:

> <quote>
> Robinson emphasized two factors in rejecting his earlier Platonism in favor of a formalist position:
>
> (i) Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless.
>
> (ii) Nevertheless, we should continue the business of Mathematics 'as usual,' i.e. we should act as if infinite totalities really existed.
> </quote>


I got the Robinson quotes from a Wikipedia article on "actual infinity". That article didn't include the second paragraph.

What Fred Jeffries wrote in a previous article is almost exactly what he should be saying in response to (ii)

[from Fred Jeffries' previous article]
> Even though you can't do something we can still obtain the result that we would get if we could do it?!

> What kind of silliness is that? How do we KNOW what we would get if we can't do it? Not only can't do it, but have no idea what it would mean to do it?

[WM]

Wo you are not willing to analyse the facts.

Enumerate all positive fractions.

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, ...
...

We check the number of indexes by bijecting them with the fractions of the first column (we could use every other column or line as well). When applying the indexes for indexing fractions such that m/n gets the index
k = (m + n - 1)(m + n - 2)/2 + m
with the resulting sequence
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...

then the integer fractions have to supply these indexes. They are stripped off these indexes. The number of not indexed fractions remains constant although all definable fractions get indexes. It remains constant in all infinitely many cases because every applied index is taken from an indexed fraction.

What is difficult to understand here?

Regards, WM

[WM]

timba...@gmail.com schrieb am Freitag, 20. Mai 2022 um 02:35:39 UTC+2:

> You've chosen to delete the relevant input of mine here. I suggest that I have instantiated a dark number and you choose to ignore it. This is not a collective situation. Once again I will put the data back in:
> 333...33, 333...34, 333...35
> are three adjacent infinite values aren't they?

No. The last digits cannot be known because they would put three dark numbers in an order. Dark numbers however cannot be ordered. This is proved best by dark fractions.

Try to enumerate all positive fractions.

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, ...
...

We check the number of available indexes by bijecting them with the fractions of the first column (we could use every other column or line as well). When applying the indexes for indexing fractions according to Cantor such that m/n gets the index

k = (m + n - 1)(m + n - 2)/2 + m
with the resulting sequence
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...

then the integer fractions have to supply these indexes. They are stripped off these indexes. The number of not indexed fractions remains constant although all definable fractions get indexes. The number of indexes remains constant in all cases because the newly applied indexes are taken from already indexed fractions.

Regards, WM

[Timothy Golden]

On Friday, May 20, 2022 at 6:08:15 AM UTC-4, WM wrote:
> timba...@gmail.com schrieb am Freitag, 20. Mai 2022 um 02:35:39 UTC+2:
>
> > You've chosen to delete the relevant input of mine here. I suggest that I have instantiated a dark number and you choose to ignore it. This is not a collective situation. Once again I will put the data back in:
> > 333...33, 333...34, 333...35
> > are three adjacent infinite values aren't they?
> No. The last digits cannot be known because they would put three dark numbers in an order. Dark numbers however cannot be ordered. This is proved best by dark fractions.

Well, perhaps you are wrong.
Here we have some infinite numbers whose last digits are specified. I know because I specified them.
I can at least increment the damn thing and I can do it many times.
I do find it interesting that you lean on the rationals. But they are not the most organized bunch.
By remaining in the modulo 10 system we can keep things simple.
It seems as though you are allowing in this format.
I guess you are saying that these are infinite numbers and that they are adjacent, but that they still are not the dark numbers that you've postulated.
I suppose that is ground I hadn't considered within the structural possibilities. I thought there was a sort of one way or the other sort of positioning that would result. You seem to have found a third rail, so to speak. I'll try to mind the gap.

>
> Try to enumerate all positive fractions.
> 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, ...
> ...
> We check the number of available indexes by bijecting them with the fractions of the first column (we could use every other column or line as well). When applying the indexes for indexing fractions according to Cantor such that m/n gets the index
> k = (m + n - 1)(m + n - 2)/2 + m
> with the resulting sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> then the integer fractions have to supply these indexes. They are stripped off these indexes. The number of not indexed fractions remains constant although all definable fractions get indexes. The number of indexes remains constant in all cases because the newly applied indexes are taken from already indexed fractions.
>
> Regards, WM

So the language of 'already indexed fractions' is ambiguous because there is tremendous redundancy. For instance in your own list 2/2 is an already indexed fraction. It's sort of like having to bring up the concept of equivalence, but it is at such a contorted level that it shouldn't have to be so twisted. The rational number somehow is in the basis of your argumentation here, as it is in mine. I find this aspect peculiar and as well have provided for the falsification of it as a fundamental value. The redundancies that your system include are a fine instance of its failings.

As you've mentioned 'collective' thinking what happens when you claim your collective to be larger than it actually is? Would you say that you have marginalized your own movement? It's like you've built in twitter bots in your list. Literally built them into your organization. If this happens to be true of twitter then their stock will fail. Markets do crash. It certainly does feel like a dark age is coming. How appropriate that the dark number could be a part of the crash. In time, at least. Fake America Fake Again.

Getting to human philosophy, which is something only the orbs can transcend; and certainly they will understand; we see the accumulation of power or control and its preservation as working elements in the academic system of mathematics. That we've all been clucking like chickens for generations... under threat of failure... it is as if anarchy could meet mathematics. I'm afraid we'll have to leave it to the orbs. We'll be following them shortly. Particularly that which the modern mathematician refuses to dismantle even though its structural content is proven to be such that can be dismantled is a failing that places him in the past. It is not so much a fraud as it is that we are at the early beginnings still. We all start as a blank slate. As to what accumulated along the way: the same old slate year after year for how many generations? Out of space yet? Just write over those older portions in a new color. We do face this sort of accumulation. It is overwhelming. I am overwhelmed by it. Further ,I find its accuracy to be dubious. That simplicity can still rule: this is just the sort of awareness that the structural argument rests upon. How odd that your natural value which I've looked askance at all these years rears its head down there. As the natural value develops the continuum in its own terms all that is needed is the secondary unity that the decimal point brings. All this business of endless digits is optional, especially when epsilon/delta shows the way.

[sergi o]

On 5/20/2022 5:03 AM, WM wrote:
> Jim Burns schrieb am Donnerstag, 19. Mai 2022 um 21:45:49 UTC+2:
>> On 5/19/2022 2:32 PM, WM wrote:
>
>>> All definable fractions get indexed.
>>> Most fractions don 't get indexed.
>>> Dark fractions.
>> All definable fractions get indexed ==
>> all fractions get indexed.
>> No dark fractions.
>
> Wo you are not willing to analyse the facts.

they are not facts at all, its your misleading bullshit

>
> Enumerate all positive fractions.
>
> 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, ...
> ...
>
> We check the number of indexes

Wrong.

by bijecting them with the fractions of the first column (we could use every other column or line as well

Wrong.

). When applying the indexes for indexing fractions such that m/n gets the index

wrong.

................................
We re-index the matrix of rationals using k

> k = (m + n - 1)(m + n - 2)/2 + m
> with the resulting sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...

done.
........................................

> then the integer fractions have to supply these indexes.

wrong.

>They are stripped off these indexes.

Wrong

>The number of not indexed fractions remains constant

Wrong.

>although all definable fractions get indexes.

wrong.

> It remains constant in all infinitely

Wrong.

> many cases because every applied index is taken from an indexed fraction.

wrong.

>
> What is difficult to understand here?

*why do you make so many intentional mistakes to justify your Darkies* ?.

>
> Regards, WM
>

[Jim Burns]

On 5/20/2022 6:03 AM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 19. Mai 2022 um 21:45:49 UTC+2:
>> On 5/19/2022 2:32 PM, WM wrote:

>>> All definable fractions get indexed.
>>> Most fractions don 't get indexed.
>>> Dark fractions.
>>
>> All definable fractions get indexed ==
>> all fractions get indexed.
>> No dark fractions.
>
> Wo you are not willing to analyse the facts.

"Calculemus." -- Gottfried Leibniz

> Enumerate all positive fractions.

These fractions:
m/n such thatm and n are _definable_ ==
only finite-many '+1' from 0 ==
for each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,

some j ends BEFORE and j+1 begins AFTER.

> 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, ...
> ...
>
> We check the number of indexes by bijecting them
> with the fractions of the first column (we could use
> every other column or line as well). When applying
> the indexes for indexing fractions such that
> m/n gets the index
> k = (m + n - 1)(m + n - 2)/2 + m

Calculemus.

For k = (m + n - 1)(m + n - 2)/2 + m

for each BEFORE and AFTER =< k,

some j ends BEFORE and j+1 begins AFTER,
because...

For each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,

some j ends BEFORE and j+1 begins AFTER.

...etc, etc, etc...

...and so,
for each BEFORE and AFTER =< (m+n-1)
and each BEFORE and AFTER =< (m+n-2),

some j ends BEFORE and j+1 begins AFTER.

For each BEFORE and AFTER =< (m+n-1)*(m+n-2),

some j ends BEFORE and j+1 begins AFTER.

because
for each BEFORE and AFTER =< (m+n-1)
and each BEFORE and AFTER =< (m+n-2),

some j ends BEFORE and j+1 begins AFTER.

and,
otherwise, there are contradictions.

| Assume OTHERWISE[1].
| Assume,
| for each BEFORE and AFTER =< (m+n-1)
| and each BEFORE and AFTER =< (m+n-2),
| some j ends BEFORE and j+1 begins AFTER,
| but
| NOT, for each BEFORE and AFTER =< (m+n-1)*(m+n-2),

| some j ends BEFORE and j+1 begins AFTER.
|

| There is a _first_ p₁+1 =< (m+n-2) such that
| NOT, for each BEFORE and AFTER =< (m+n-1)*(p₁+1),
| some j ends BEFORE and j+1 begins AFTER,
| AND it is
| TRUE, for each BEFORE and AFTER =< (m+n-1)*p₁,

| some j ends BEFORE and j+1 begins AFTER.
|

| By definition of '*',
| (m+n-1)*(p₁+1) = (m+n-1)*p₁+(m+n-1)
|
| However,
| for each BEFORE and AFTER =< (m+n-1)*p₁+(m+n-1),
| some j ends BEFORE and j+1 begins AFTER,
| because
| for each BEFORE and AFTER =< (m+n-1)*p₁,
| for each BEFORE and AFTER =< (m+n-1),
| some j ends BEFORE and j+1 begins AFTER,
| and,
| otherwise, there are contradictions.
|
|| Assume OTHERWISE[2].
|| Assume,
|| for each BEFORE and AFTER =< (m+n-1)*p₁
|| and each BEFORE and AFTER =< (m+n-1),
|| some j ends BEFORE and j+1 begins AFTER,
|| but
|| NOT, for each BEFORE and AFTER =< (m+n-1)*p₁+(m+n-1),

|| some j ends BEFORE and j+1 begins AFTER.
||

|| There is a _first_ s₁+1 =< (m+n-1) such that
|| NOT, for each BEFORE and AFTER =< (m+n-1)*p₁+(s₁+1),
|| some j ends BEFORE and j+1 begins AFTER,
|| AND it is
|| TRUE, for each BEFORE and AFTER =< (m+n-1)*p₁+s₁,

|| some j ends BEFORE and j+1 begins AFTER.
||

|| By definition of '+',
|| (m+n-1)*p₁+(s₁+1) = ((m+n-1)*p₁+s₁)+1
||
|| However,
|| for each BEFORE and AFTER =< ((m+n-1)*p₁+s₁)+1
|| some j ends BEFORE and j+1 begins AFTER,
|| because
|| for each BEFOREless and AFTERless =< (m+n-1)*p₁+s₁,
|| some j ends BEFOREless and j+1 begins AFTERless
||
|| For each BEFORE and AFTER =< ((m+n-1)*p₁+s₁)+1
||
|| either (i)
|| BEFORE = BEFOREless
|| AFTER = AFTERless∪{((m+n-1)*p₁+s₁)+1}
|| and the j which ends BEFOREless ends BEFORE
|| and the j+1 which begins AFTERless begins AFTER
||
|| or (ii)
|| BEFORE = {all =< (m+n-1)*p₁+s₁}
|| AFTER = {((m+n-1)*p₁+s₁)+1}
|| and the j which ends BEFORE = (m+n-1)*p₁+s₁
|| and the j+1 which begins AFTER = ((m+n-1)*p₁+s₁)+1
||
|| CONTRADICTION[2]:
|| NOT, for each BEFORE and AFTER =< (m+n-1)*p₁+(s₁+1),
|| some j ends BEFORE and j+1 begins AFTER,
|| AND it is
|| TRUE, for each BEFORE and AFTER =< (m+n-1)*p₁+(s₁+1),

|| some j ends BEFORE and j+1 begins AFTER.
|

| Therefore,
| for each BEFORE and AFTER =< (m+n-1)*p₁+(m+n-1),

| some j ends BEFORE and j+1 begins AFTER.

| and,
| for each BEFORE and AFTER =< (m+n-1)*(p₁+1),

| some j ends BEFORE and j+1 begins AFTER.
|

| CONTRADICTION[1]:
| NOT, for each BEFORE and AFTER =< (m+n-1)*(p₁+1),
| some j ends BEFORE and j+1 begins AFTER,
| AND it is
| TRUE, for each BEFORE and AFTER =< (m+n-1)*(p₁+1),

| some j ends BEFORE and j+1 begins AFTER.

Therefore,
for each BEFORE and AFTER =< (m+n-1)*(m+n-2),

some j ends BEFORE and j+1 begins AFTER.

...etc, etc, etc...

Therefore,
for k = (m + n - 1)(m + n - 2)/2 + m

for each BEFORE and AFTER =< k,
some j ends BEFORE and j+1 begins AFTER.

Therefore,
k/1 is one of "these fractions",
k is _definable_
k is only finite-many '+1' from 0

> with the resulting sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> then the integer fractions have to supply these indexes.

...which we just now saw that they do.

> They are stripped off these indexes.
> The number of not indexed fractions remains constant

For each indexed fraction,
the number of _indexed_ fractions after it
remains constant and infinite.

For k = (m + n - 1)(m + n - 2)/2 + m
there are no _not-indexed_ fractions.

Infinity is not a reallyreallyreallyreallyreallyreally
large number. It is a different kind of thing.

> although all definable fractions get indexes.

All definable fractions == all fractions.
All fractions get indexes.

> It remains constant in all infinitely many cases
> because every applied index is taken from
> an indexed fraction.
>
> What is difficult to understand here?

My best guess is that you don't understand that
infinity is not a reallyreallyreallyreallyreallyreally
large number. It is a different kind of thing.

[WM]

timba...@gmail.com schrieb am Freitag, 20. Mai 2022 um 14:02:10 UTC+2:
> On Friday, May 20, 2022 at 6:08:15 AM UTC-4, WM wrote:

> Here we have some infinite numbers whose last digits are specified.

But the number of their digits is not specified.

> > Try to enumerate all positive fractions.
> > 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, ...
> > ...
> > We check the number of available indexes by bijecting them with the fractions of the first column (we could use every other column or line as well). When applying the indexes for indexing fractions according to Cantor such that m/n gets the index
> > k = (m + n - 1)(m + n - 2)/2 + m
> > with the resulting sequence
> > 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> > then the integer fractions have to supply these indexes. They are stripped off these indexes. The number of not indexed fractions remains constant although all definable fractions get indexes. The number of indexes remains constant in all cases because the newly applied indexes are taken from already indexed fractions.
> >

> So the language of 'already indexed fractions' is ambiguous because there is tremendous redundancy. For instance in your own list 2/2 is an already indexed fraction.

No, the fractions 1/1 and 2/2 have the same value but are different fractions.

Regards, WM

[WM]

But these k don't cover all |N because they have aleph_0 successors in |N.

>
> Infinity is not a reallyreallyreallyreallyreallyreally
> large number. It is a different kind of thing.

That is why your k are irrelevant as a vanishing minority.

> > every applied index is taken from
> > an indexed fraction.
> >
> > What is difficult to understand here?
> My best guess is that you don't understand that
> infinity is not a reallyreallyreallyreallyreallyreally
> large number. It is a different kind of thing.

When we take an element and throw it away, then we will not collect a set. Do you believe that this will change in infinity?

Regards, WM

[sergi o]

there is infinite redundancy

>
> No, the fractions 1/1 and 2/2 have the same value but are different fractions.
>
> Regards, WM
>

WM remains confused, from above;

integer fractions
indexed fractions
definable fractions
already indexed fractions

[Timothy Golden]

big snip of lots of BEFORE and AFTER

> For each indexed fraction,
> the number of _indexed_ fractions after it
> remains constant and infinite.
>
> For k = (m + n - 1)(m + n - 2)/2 + m
> there are no _not-indexed_ fractions.
>
> Infinity is not a reallyreallyreallyreallyreallyreally
> large number. It is a different kind of thing.
> > although all definable fractions get indexes.
> All definable fractions == all fractions.
> All fractions get indexes.

In this parlor of the infinite the elaborate redundancies that are being worked on get wiped away by some large value under whose rational system all values up though n can be enlisted via multiples of 1/n!.

Then too, this is overgrown since the halves are already covered by the quarters and so the critical value is much smaller than the factorial.
You all would have a better sense of what this critical value is, and it is larger by far than n, but in its simple index all of those priors will be present.
Could this value somehow be of interest? How does it go?
1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720,
360360, 360360, 360360, 720720, 12252240 (at 17 now),
12252240, 232792560 (19), 232792560, 232792560,
232792560, 5354228880 (23), ...
It's not pretty. It's more obtuse, but it is another way.

I'm still waiting for more to weigh in on the legitimacy of the large value:
333...35
as a working value. Sure it's large; infinite in nature, but doesn't it help knowing what the last few digits are?
Of course it does. They are critical to the mechanics of computation. Without them the notion of a successor is not going to happen.
I think it is absurd, yet it is as well workable.
It seems according to WM that by instantiating this instance I have sunk his own theory of dark numbers.
I hesitate to make this claim too strongly, and there are bizarre instances of things to do with these large values.
Still there are at least eight of them. That this is a result of the numerical base that we work in means that if you needed more you could make some more as well by cracking that open.

Is it interesting that the carry operation becomes significant in this analysis? Here we see the action of the digits making consequences on the workable values. This is good, but in this depth the question of whether our digital values have operations embedded within them has to be refreshed. It looks as if this carry operator arguable does exist, but this does not legitimate any claim that division exists. Operators and values seem to be distinct concepts by the works of humans. Linguistically this is close by to Chomsky's analysis of language possibly. These are nouns and verbs of the simplest form. As to what it means to have a 'he' who cannot be named specifically... this is problematic. This is the puzzle of instantiation which often enough evaporates the works of mathematicians who are stuck in the abstracted state. That a working instance could evaporate their work is entirely feasible.

Could it be said that these infinite values refuse to multiply? Would that be a problem?
Or is it even fair to vary the head and the tail uniquely so that
567333...36
is as well a valid infinite form? Leaning back on those pesky rationals they do happily instance things such as
1.121212...
and so I guess reusing their logic without any vindictive prejudice we should then allow in
1121212...1216
as a valid infinite value, no?
So then there is no shortage even in base ten.
This would be the permissive strategy I suppose.
Every digit is specified and counts. Therefor these are the best defined numbers in existence.

Sincerely I would think you will have to falsify these or accept them. Each choice is consequential.
Bifurcation.

For me the logic will extend back onto the rationals as they are taught with their digital ellipses.
This is a structural argument. The decimal value is the natural value augmented with a little dot.

[Fritz Feldhase]

On Saturday, May 21, 2022 at 5:30:17 PM UTC+2, WM wrote:
> Jim Burns schrieb am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:
> >
> > k = (m + n - 1)(m + n - 2)/2 + m
> > there are no _not-indexed_ fractions.
> >

> But these k don't cover all IN because they have aleph_0 successors in IN.

I see. So let's consider the/all k with k e IN.

So "these k don't cover all IN because they have aleph_0 successors in IN."

(Actually, "these k don't cover all IN because _each and every_ k has aleph_0 successors in IN.")

Yeah, exactly!

I'd like to call this fact the Hauptsatz of Mückenmath!

[Jim Burns]

On 5/21/2022 11:30 AM, WM wrote:
> Jim Burns schrieb
> am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:

>> For k = (m + n - 1)(m + n - 2)/2 + m
>> there are no _not-indexed_ fractions.
>
> But these k don't cover all |N because
> they have aleph_0 successors in |N.

Describe these fractions m/n.

Describe these k/1 as some, but not all, of the m/n.

From the descriptions, we know that

for each k, there is m/n such that
k = (m+n-1)*(m+n-2)/2+m

and
for each m/n, there is k such that
k = (m+n-1)*(m+n-2)/2+m

The matrix of m/n:k/1

1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ...
2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ...
3/1:6/1 3/2:9/1 3/3:13/1 3/4:18/1 ...
4/1:10/1 4/2:14/1 4/3:19/1 4/4:25/1 ...
5/1:15/1 5/2:20/1 5/3:26/1 5/4:33/1 ...
6/1:21/1 6/2:27/1 6/3:34/1 6/4:42/1 ...
7/1:28/1 7/2:35/1 7/3:43/1 7/4:52/1 ...
8/1:36/1 8/2:44/1 8/3:53/1 8/4:63/1 ...
9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ...
... ... ... ...

>> Infinity is not a reallyreallyreallyreallyreallyreally
>> large number. It is a different kind of thing.
>
> That is why your k are irrelevant as a
> vanishing minority.

The relevance of this whole argument is that
the k are a vanishing minority which are
nonetheless enough to match the whole matrix.

Your counter-argument, if one calls it that, is that
this obviously can't be true, and, therefore,
it isn't true.

Your judgment of obvious-falsehood is rooted in
your experience of _some_ collections. Sheep.
Pebbles. Sand grains. Stars.

That sheep-pebbles-sand-stars experience is not
representative of all collections. Some collections
can be matched by a vanishing minority. This is
the point being made by k = (m+n-1)*(m+n-2)/2+m

[Timothy Golden]

That's two very weak points, Wolfgang.
Could it be that infinite values are capable of having specific successors and predecessors while their specific ordering is not actually possible?
In other words is it naive to claim that
111...1113 < 222...2
In effect each unique head has a working tail but the uniqueness of the head places it in its own isolated space? This is sort of a dimensional argument, actually. Stating that
111...13 + 1 = 111...14
is not controversial is it? As you claim to be able to discriminate equivalent values perhaps you would care to ponder:
111...13 =?= 111...13
and I see no controversy here. This is a unique representation. It is as well readily evaluable I think as to whether
10111...13 =?= 111...13
where clearly the answer is no, but now as to which value is the lesser: this is why I argue that the heads of these numbers are isolated. We are somewhat establishing some form of dimensionality. Still their tails are operable. I guess it wouldn't be long to start cutting up such values and calling them multidimensional, which will then get us into another ellipses at our whim, just as the first occurred.

Certainly the form is demanding redundancy of digits in the middle otherwise no compaction can occur, which the ellipses develop. In effect all these infinite forms suffer this informational ambiguity... unless you try to let in the irrational instances... something I think I'll try not to consider here.
I think philosophically it would be great to develop a system that denies many usages of the ellipses and treats them as an abusive form that no compiler or procedure can actually handle; under this guise every usage of them is a bug within mathematical thinking. They are used in so many places that I wouldn't care to claim this universally. I do think that this level of discussion is possible within this topic. It is especially thanks to your usage of the rational values that these sorts of things come into being. Again, back in the establishment of the real number it is epsilon/delta that relaxes things. This is a procedure which occurs at the tail of numbers, but not at the tail of natural numbers. No: perfection is intact on these infinite forms. Please tell me a digit that is not specified, and to confront your first fib what infinite value should have finitely many digits? None of them, sir. You have to be engaged in this format.

[sergi o]

On 5/21/2022 10:30 AM, WM wrote:
> Jim Burns schrieb am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:
>> On 5/20/2022 6:03 AM, WM wrote:
>
>> For k = (m + n - 1)(m + n - 2)/2 + m
>> there are no _not-indexed_ fractions.
>
> But these k don't cover all |N because they have aleph_0 successors in |N.

wrong!

For k = (m + n - 1)(m + n - 2)/2 + m holds for all m,n and k in |N

"because they have aleph_0 successors in |N" shows *you do not understand equations*!

>>
>> Infinity is not a reallyreallyreallyreallyreallyreally
>> large number. It is a different kind of thing.
>
> That is why your k are irrelevant as a vanishing minority.

you have lost the argument, and are irrelevant.

>
>>> every applied index is taken from
>>> an indexed fraction.
>>>
>>> What is difficult to understand here?
>> My best guess is that you don't understand that
>> infinity is not a reallyreallyreallyreallyreallyreally
>> large number. It is a different kind of thing.
>
> When we take an element and throw it away, then we will not collect a set.

when you take a sheep out of a set of sheeps, you have a different set of sheeps.

you will not collect the set of sheeps, until you pay for them.

> Do you believe that this will change in infinity?

you have no concept of infinity

>
> Regards, WM

[Ross A. Finlayson]

On Wednesday, May 18, 2022 at 11:54:33 AM UTC-7, timba...@gmail.com wrote:
> On Wednesday, May 18, 2022 at 2:32:16 PM UTC-4, WM wrote:

> > FredJeffries schrieb am Mittwoch, 18. Mai 2022 um 18:14:16 UTC+2:
> > >
> > > <quote>
> > > Robinson emphasized two factors in rejecting his earlier Platonism in favor of a formalist position:
> > >
> > > (i) Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless.
> > >
> > > (ii) Nevertheless, we should continue the business of Mathematics 'as usual,' i.e. we should act as if infinite totalities really existed.
> > > </quote>

> > Let's continue with the origianl source:
> >
> > "I feel quite unable to grasp the idea of an actual infinite totality. To me there appears to exist an unbridgeable gulf between sets or structures of one, or two, or five elements, on one hand, and infinite structures on the other hand [...] I must regard a theory which refers to an infinite totality as meaningless in the sense that its terms and sentences cannot posses the direct interpretation in an actual structure that we should expect them to have by analogy with concrete (e.g., empirical) situations." [A. Robinson: "Formalism 64", North-Holland, Amsterdam, p. 230f]
> >
> > Regards, WM
> I wonder to what degree those concerned with natural valued infinity are possibly entertaining the birth of the continuum?
> As you start plopping down positions on a line and they keep going at some large value the discernment of the smaller values seems imperceptible. Particularly thinking in terms of large n a relative position abstraction with arbitrarily fine granularity ensues.
>
> This bears out as we consider the decimal value as a natural value. For instance as 1/3= 0.333... then dropping the decimal point we are dealing in a natural value 333.... Even a value such as 2/5 in perfection will yield 0.4000... which again as a natural value is 4000...
> These sorts of infinite precision values are in denial of epsilon/delta theory, whereas 2/5=0.4000 is a finite precision instance. In other words this is a computationally valid instance and if we did wish to work in greater precision we could. Dropping the decimal from this instance we see 2/5 as 4000 and of course its data can be recovered by building up a structure xe where x is the natural value and e is the decimal place. The 'e' portion is arguably natural valued, but it is of a different meaning. It's position is a matter of placing a secondary unity upon the otherwise purely natural value.
>
> In this way the continuum can be built by having a high regard for the natural value; augmenting it with a new sense of unity; and as well keeping a regard for epsilon/delta theory and its adjustable precision. In effect those presuming perfection in their rational values have been working with an infinite form of the natural value and never saw it as ambiguous. Could these be the infinite concerns of the natural valued philosophy? I do credit in part WM's dark number with helping me develop the gray number that is the continuum; the more truly 'real' value. That the interpretation could come back then onto the naturals in this way is an interesting turn. So then are values such as 333... dark values? Hah! I have managed to instantiate them!

"Double or nothing"

It is like physics with the relativist and absolutist again,
simply mathematics with the regular iteration and regular increment, again.

Then, it really is at least _some_ matters of philosophy,
that the primary objects of mathematics are defined in
a usual language of a philosophy of mathematics, because
mathematical objects fulfill the definition of being mathematical objects.

So, while there are paradoxes like Cantor's paradox or Kunen's paradox,
or Russell and Burali-Forti, again here are not (paradoxes) , being explained
with mathematical logic, why there is the extra-ordinary, and it's quite
well-phrased already in the usual most technical philosophy,
for the usual fundamental canon (or inventing one).

Robinson's "infinitesimals" are basically just only the neighborhoods
about a point. The "hyper-integers" or "hyper-reals", are not much more
than that after Skolem, having a "non-standard, countable ... model" or
"standard, countable middle", the integers, those are Skolem's also.


Peano for example has ideas like "these standard infinitesimals are
like Leibniz' differentials, as far as I'm concerned" - correctly framed.

[WM]

Fritz Feldhase schrieb am Samstag, 21. Mai 2022 um 18:25:13 UTC+2:
> On Saturday, May 21, 2022 at 5:30:17 PM UTC+2, WM wrote:
> > Jim Burns schrieb am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:
> > >
> > > k = (m + n - 1)(m + n - 2)/2 + m
> > > there are no _not-indexed_ fractions.
> > >
> > But these k don't cover all IN because they have aleph_0 successors in IN.
>
> I see. So let's consider the/all k with k e IN.
>
> So "these k don't cover all IN because they have aleph_0 successors in IN."
>
> (Actually, "these k don't cover all IN because _each and every_ k has aleph_0 successors in IN.")

Here you can learn what the k c an do:

If the fractions m/n are enumerated by the natural numbers k according to Cantor's function

k = (m + n - 1)(m + n - 2)/2 + m

then all the fractions of the sequence

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

are enumerated.

But if the natural numbers first are in bijection with the integer fractions of the first column of the matrix

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, ...
...

then they must be distributed over the matrix such that no fraction remains without index. That means, there is a permutation such that the X of the first column

XOOOO...
XOOOO...
XOOOO...
XOOOO...
XOOOO...
...

by being exchanged with the O's cover all matrix positions. All O's will vanish. This is obviously impossible because exchanging cannot reduce them. The number of not indexed fractions, represented by O's, will remain constant forever, in infinity.

Regards, WM

[WM]

Jim Burns schrieb am Samstag, 21. Mai 2022 um 19:06:27 UTC+2:
> On 5/21/2022 11:30 AM, WM wrote:

> > That is why your k are irrelevant as a
> > vanishing minority.
> The relevance of this whole argument is that
> the k are a vanishing minority which are
> nonetheless enough to match the whole matrix.

The k do not cover the matrix but only one column at the start. During the infinite operation their share cannot grow .

>
> Your counter-argument, if one calls it that, is that
> this obviously can't be true, and, therefore,
> it isn't true.
>
> Your judgment of obvious-falsehood is rooted in
> your experience of _some_ collections.

My judgement is rooted in the fact that an infinite sequence of +1, -1, +1, -1, +1, -1, ... can never, at no finite step and not in the limit either, lead to value larger than 10.

> That sheep-pebbles-sand-stars experience is not
> representative of all collections. Some collections
> can be matched by a vanishing minority. This is
> the point being made by k = (m+n-1)*(m+n-2)/2+m

For that assumption you have to claim that the sequence +1, -1, +1, -1, +1, -1, ... will acumulate infinitely many units. I will not join you. Hardly will any mathematician unless he is told that otherwise set theory would be wrong.

Regards, WM

[Fritz Feldhase]

On Sunday, May 22, 2022 at 3:32:46 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Samstag, 21. Mai 2022 um 18:25:13 UTC+2:
> > On Saturday, May 21, 2022 at 5:30:17 PM UTC+2, WM wrote:
> > > Jim Burns schrieb am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:
> > > >
> > > > k = (m + n - 1)(m + n - 2)/2 + m
> > > > there are no _not-indexed_ fractions.
> > > >
> > > But these k don't cover all IN because they have aleph_0 successors in IN.
> >
> > I see. So let's consider the/all k with k e IN.
> >
> > So "these k don't cover all IN because they have aleph_0 successors in IN."
> >
> > (Actually, "these k don't cover all IN because _each and every_ k has aleph_0 successors in IN.")

No reaction/answer?

Ok. So let's consider the following:

> If the fractions m/n are enumerated by the natural numbers k according to Cantor's function
> k = (m + n - 1)(m + n - 2)/2 + m
> then all the fractions of the sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> are enumerated.
>

> [I]f the natural numbers [...] are in bijection with the integer fractions of the first column of the matrix

>
> 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, ...
> ...
>

> then they [can] be distributed over the matrix such that no fraction remains without index. That means, there is a [distribution] such that the X of the first column

>
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> ...
>

> [...] cover all matrix positions.

Indeed!

There's a simple distribution of the X to achieve this goal:

> 1/1, 1/2, 1/4, 1/7, ...
> 1/3, 1/5, 1/8, ...
> 1/6, 1/9, ...
> 1/10, ...
> ...

Well done, Mückenheim!

[Fritz Feldhase]

On Sunday, May 22, 2022 at 3:42:20 PM UTC+2, WM wrote:

> My judgement is rooted in the fact that an infinite sequence of +1, -1, +1, -1, +1, -1, ... can never, at no finite step and not in the limit either

This sequence DOES NOT HAVE a limit, you silly idiot!

> lead to value larger than 10.

Since all terms in this sequence are either 1 or -1, no term is > 10. WOW!

An incredibe insight, Mückenheim!

[Timothy Golden]

On Saturday, May 21, 2022 at 5:18:00 PM UTC-4, Ross A. Finlayson wrote:
> On Wednesday, May 18, 2022 at 11:54:33 AM UTC-7, timba...@gmail.com wrote:
> > On Wednesday, May 18, 2022 at 2:32:16 PM UTC-4, WM wrote:
> > > FredJeffries schrieb am Mittwoch, 18. Mai 2022 um 18:14:16 UTC+2:
> > > >
> > > > <quote>
> > > > Robinson emphasized two factors in rejecting his earlier Platonism in favor of a formalist position:
> > > >
> > > > (i) Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless.
> > > >
> > > > (ii) Nevertheless, we should continue the business of Mathematics 'as usual,' i.e. we should act as if infinite totalities really existed.
> > > > </quote>
> > > Let's continue with the origianl source:
> > >
> > > "I feel quite unable to grasp the idea of an actual infinite totality. To me there appears to exist an unbridgeable gulf between sets or structures of one, or two, or five elements, on one hand, and infinite structures on the other hand [...] I must regard a theory which refers to an infinite totality as meaningless in the sense that its terms and sentences cannot posses the direct interpretation in an actual structure that we should expect them to have by analogy with concrete (e.g., empirical) situations." [A. Robinson: "Formalism 64", North-Holland, Amsterdam, p. 230f]
> > >
> > > Regards, WM
> > I wonder to what degree those concerned with natural valued infinity are possibly entertaining the birth of the continuum?
> > As you start plopping down positions on a line and they keep going at some large value the discernment of the smaller values seems imperceptible. Particularly thinking in terms of large n a relative position abstraction with arbitrarily fine granularity ensues.
> >
> > This bears out as we consider the decimal value as a natural value. For instance as 1/3= 0.333... then dropping the decimal point we are dealing in a natural value 333.... Even a value such as 2/5 in perfection will yield 0.4000... which again as a natural value is 4000...
> > These sorts of infinite precision values are in denial of epsilon/delta theory, whereas 2/5=0.4000 is a finite precision instance. In other words this is a computationally valid instance and if we did wish to work in greater precision we could. Dropping the decimal from this instance we see 2/5 as 4000 and of course its data can be recovered by building up a structure xe where x is the natural value and e is the decimal place. The 'e' portion is arguably natural valued, but it is of a different meaning. It's position is a matter of placing a secondary unity upon the otherwise purely natural value.
> >
> > In this way the continuum can be built by having a high regard for the natural value; augmenting it with a new sense of unity; and as well keeping a regard for epsilon/delta theory and its adjustable precision. In effect those presuming perfection in their rational values have been working with an infinite form of the natural value and never saw it as ambiguous. Could these be the infinite concerns of the natural valued philosophy? I do credit in part WM's dark number with helping me develop the gray number that is the continuum; the more truly 'real' value. That the interpretation could come back then onto the naturals in this way is an interesting turn. So then are values such as 333... dark values? Hah! I have managed to instantiate them!
>
> "Double or nothing"
>
> It is like physics with the relativist and absolutist again,
> simply mathematics with the regular iteration and regular increment, again.
>
> Then, it really is at least _some_ matters of philosophy,
> that the primary objects of mathematics are defined in
> a usual language of a philosophy of mathematics, because
> mathematical objects fulfill the definition of being mathematical objects.

Yes, well here I think that as you distinguish mathematical objects from philosophy you are implying the existence of some pure form with a mechanism that can be detailed which deserves attention. As to when such a thing becomes 'mathematical'; isn't it a matter of ambiguity? To achieve 'mathematical' implies purity; free of ambiguity. So long as we discover ambiguity within the constructions then this terminology suffers.

I don't generally care to dabble with infinity as a touchable thing. If it doesn't work at ten thousand and work better at a hundred thousand it's probably not going to work at all.

Then too there was a time when these branches were better joined; when philosophy and mathematics and physics all mattered to one mind. This classical mindset is nearly gone. Lopping these branches off from one another is suicidal. It would be wiser for the quantum people to admit that they seek a new philosophy rather than claim immunity from philosophy altogether. Will there be any consequence when the quantum computer never arrives? Never say never... The matter of time shows that we are engaged in a progression but also in the extreme accumulation that we are facing the hope of great simplification should be kept alive. Otherwise we are buried.

>
> So, while there are paradoxes like Cantor's paradox or Kunen's paradox,
> or Russell and Burali-Forti, again here are not (paradoxes) , being explained
> with mathematical logic, why there is the extra-ordinary, and it's quite
> well-phrased already in the usual most technical philosophy,
> for the usual fundamental canon (or inventing one).
>
> Robinson's "infinitesimals" are basically just only the neighborhoods
> about a point. The "hyper-integers" or "hyper-reals", are not much more
> than that after Skolem, having a "non-standard, countable ... model" or
> "standard, countable middle", the integers, those are Skolem's also.

Not so well versed in these people, but how about Dedekinds irrational system applied to all on the continuum?
This is the gray number. This is the engineer's number. The physicists number. Sad that I cannot say that this is the real number.

[Jim Burns]

On 5/22/2022 9:42 AM, WM wrote:
> Jim Burns schrieb
> am Samstag, 21. Mai 2022 um 19:06:27 UTC+2:

>> That sheep-pebbles-sand-stars experience is not
>> representative of all collections. Some collections
>> can be matched by a vanishing minority. This is
>> the point being made by k = (m+n-1)*(m+n-2)/2+m
>
> For that assumption

I describe a fraction m/n.
That's the assumption:
That the fractions we are talking about are
as I describe them.

_From that description_ I show that k/1
is a vanishing minority of those fractions.
That part is NOT an assumption.

For each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,
some j ends BEFORE and j+1 begins AFTER.

From that description of each BEFORE and AFTER,
we know that,
if
there are _any_ m and n for which
it's false that,
for each BEFORE and AFTER =< m+n,

some j ends BEFORE and j+1 begins AFTER,

then
there is a _first_ n₁+1 =< n for which
it's false that,
for each BEFORE and AFTER =< m+(n₁+1),

some j ends BEFORE and j+1 begins AFTER.

Because n₁+1 is _first_
we know that
it's true that,
for each BEFORE and AFTER =< m+n₁,

some j ends BEFORE and j+1 begins AFTER.

That leads to a contradiction.
Therefore,
there is no _first_ n₁+1 for which that's false.
Therefore,
there is no n _at all_ for which that's false.
Therefore,
for each BEFORE and AFTER =< m+n,

some j ends BEFORE and j+1 begins AFTER.

And so on. The same argument gets repeated with
minor variations.

> For that assumption you have to claim that the
> sequence +1, -1, +1, -1, +1, -1, ... will acumulate
> infinitely many units. I will not join you. Hardly
> will any mathematician unless he is told that
> otherwise set theory would be wrong.

If,

for each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,

some j ends BEFORE and j+1 begins AFTER,
and
k = (m+n-1)*(m+n-2)/2+m
then

[Fritz Feldhase]

Ooops, I meant

> > 1/1, 2/1, 4/1, 7/1, ...
> > 3/1, 5/1, 8/1, ...
> > 6/1, 9/1, ...

[Jim Burns]

On 5/22/2022 9:32 AM, WM wrote:
> Fritz Feldhase schrieb
> am Samstag, 21. Mai 2022 um 18:25:13 UTC+2:

>> [...]

> If the fractions m/n are enumerated by
> the natural numbers k according to Cantor's function
> k = (m + n - 1)(m + n - 2)/2 + m
> then all the fractions of the sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> are enumerated.
>
> But if the natural numbers first are in bijection with
> the integer fractions of the first column of the matrix

... of m/n:k/1

1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ...
2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ...
3/1:6/1 3/2:9/1 3/3:13/1 3/4:18/1 ...
4/1:10/1 4/2:14/1 4/3:19/1 4/4:25/1 ...
5/1:15/1 5/2:20/1 5/3:26/1 5/4:33/1 ...
6/1:21/1 6/2:27/1 6/3:34/1 6/4:42/1 ...
7/1:28/1 7/2:35/1 7/3:43/1 7/4:52/1 ...
8/1:36/1 8/2:44/1 8/3:53/1 8/4:63/1 ...
9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ...

.. ... ... ...

> then they must be distributed over the matrix
> such that no fraction remains without index.

Note that no fraction is without index.

> That means, there is a permutation such that
> the X of the first column
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> ...
>
> by being exchanged with the O's cover all
> matrix positions.

Yes.

> All O's will vanish.

Yes.

> This is obviously impossible

No. See above.

> This is obviously impossible because exchanging
> cannot reduce them.

Exchanging does not reduce them.
They are not reduced.
They are the same size (cardinality) as
one of their proper subsets.
See above.

Some collections can be the same size (cardinality)
as one of their proper subsets.
For example, see above.

> The number of not indexed fractions,

No fraction is without index.
See above.

[zelos...@gmail.com]

torsdag 19 maj 2022 kl. 15:44:32 UTC+2 skrev WM:
> Jim Burns schrieb am Donnerstag, 19. Mai 2022 um 02:30:59 UTC+2:
> > On 5/18/2022 4:48 PM, WM wrote:
>
> > > Between every number subject to induction and omega
> > > there are aleph_0 natural numbers which are not
> > > subject to induction because they do never vanish.
> > You are facing in the wrong direction.
> So it appears to someone who never faced in the right direction. What is wrong with my direction? (Except that it disproves your beliefs.)
> > Stand on a number k finitely-many '+1" from 0.
> > Turn away from omega and look at 0.
> >
> > k is subject to induction *because*
> > k is finitely-many '+1' from 0.

> That does not exclude aleph_0 dark numbers immediately before omega.

All of which you fail to define in any meaningful way.

> >
> > Infinity enters the story because there are
> > infinitely-many numbers which are finitely-many '+1'
> > from 0.
> Impossible if actual infinity is meant, that is *more* than any finite number of +1.
>
> Every finite number of +1 is contained in a potentially infinite set which is followed by an infinite set - if infinity is actual.

there is no "potential infinite set", that is not a defined thing in mathematics.

>
> Regards, WM

[WM]

Please explain why the sequence -1 + 1 - 1 + 1 - 1 + 1 -+ has sum oo or write down the complete matrix in order to prove your claim.

Regards, WM

[WM]

Fritz Feldhase schrieb am Sonntag, 22. Mai 2022 um 17:19:30 UTC+2:
> On Sunday, May 22, 2022 at 3:42:20 PM UTC+2, WM wrote:
>
> > My judgement is rooted in the fact that an infinite sequence of +1, -1, +1, -1, +1, -1, ... can never, at no finite step and not in the limit either

> Since all terms in this sequence are either 1 or -1, no term is > 10. WOW!
>

This sequence yields the share of fractions enumerated in addition to the outset.

Regards, WM

[WM]

There is no relevance of BEFORE and AFTER. There is relevance only of the fact that every indexed fraction can get an index only from a previously indexd fraction. And previously most fractions were not indexed.

Regards, WM

[WM]

Jim Burns schrieb am Sonntag, 22. Mai 2022 um 20:49:56 UTC+2:
> On 5/22/2022 9:32 AM, WM wrote:

> > That means, there is a permutation such that
> > the X of the first column
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > ...
> >
> > by being exchanged with the O's cover all
> > matrix positions.
> Yes.

By exchanging X and O the shares will remain constant.
I am not willing to discuss with stubborn stupidity.
Come back when you will have recovered.
EOD.

Regards, WM

[WM]

zelos...@gmail.com schrieb am Montag, 23. Mai 2022 um 07:06:09 UTC+2:
> torsdag 19 maj 2022 kl. 15:44:32 UTC+2 skrev WM:

> > > k is subject to induction *because*
> > > k is finitely-many '+1' from 0.
> > That does not exclude aleph_0 dark numbers immediately before omega.
> All of which you fail to define in any meaningful way.

All natnumbers which you can define have an infinite distance from omega. This distance is existing but undefinable.

Regards, WM

[FromTheRafters]

WM was thinking very hard :

> Jim Burns schrieb am Sonntag, 22. Mai 2022 um 20:49:56 UTC+2:
>> On 5/22/2022 9:32 AM, WM wrote:
>
>>> That means, there is a permutation such that
>>> the X of the first column
>>> XOOOO...
>>> XOOOO...
>>> XOOOO...
>>> XOOOO...
>>> XOOOO...
>>> ...
>>>
>>> by being exchanged with the O's cover all
>>> matrix positions.
>> Yes.
>
> By exchanging X and O the shares will remain constant.
> I am not willing to discuss with stubborn stupidity.

Decades prove otherwise.

[Timothy Golden]

Doesn't the algebraic index convert back to a one dimensional system?
Is there some consequence to requiring a two dimensional format?
Meanwhile we all must admit that there is a clean ordering of these supposed continuous values that fits them on a line, and this perfect order cannot be attained by your works. This is because the finest gradation is at large value (denominator). Where we do discover clean order is simply choosing a large denominator and exposing how many of these early values can be covered by it. The willingness to play around with infinities here suggests you are comfortable specifying this value. Is it merely a matter of specifying this odd algorithm and enlisting induction through it? So for any d you choose I can fit it and all predecessors on some large base b. Spec the algorithm, and invoke induction as d+1?

Nah. I've got a better way. Same old method used on some past thread. Each line is worked modulo n so the first fraction line becomes
0.1, 0.1, 0.1, 0.1, 0.1, ...
where the first is modulo-one, the second is modulo-two, and so forth. The next line is just twice the first line
Each line is merely the sum of this first course with the previous course. The nth line is simply the sum from 1 to n of 0.1 with some subscript say to denote the modulo nature, but we now have a simplified expression. In effect the entire construction collapses to a constant notation.
Sigma of 0.1 basically.

No idea how this format is convincing to anybody. What matters to me is to expose that the decimal value as in with the usage of the decimal point is in a class of its own. The claim that it is a rational value is a fraud that is gotten by building the rational value first and calling it fundamental. It is not. It contains an operator and therefor is not fundamental. Upon computation we generally land with the decimal point usage, and it's structure stands out as a new format of number. The disregard of the floating point value by the mathematicians is probably the second worst misnomer that status quo mathematics sits upon; the first being sign.

As the rational value in general is exposed as a reradixed system you all have to be labelled as dirty reradixers. That epsilon/delta theory has something to say here, and particularly when applied universally to the floating point value, then a different interpretation of number ensues. The most stubborn point in it all is that the magic and supremacy of your natural value as fundamental actually returns into the continuum. It is just that the continuum is gray, whereas the naturals are black on white. The en masse delusion is secure obviously and has been far longer than formalizations of the real number have been around. Indeed, as to which came first arguably it was the continuum; and here maybe lays some deep physical correspondence that we don't have yet. In effect mathematical theory does ask for spacetime correspondence, for all of our symbolic works have been carried out in a medium and via techniques whose side-effects may be consequential. It is this draw down to a blank slate and its early beginnings where simplicity lays. She barely awoke and the accumulation began. It might as well be true that our intelligences are actually diminishing as modern society has polluted us with lead and so forth. Perhaps the coming generation will be better off; some of them hopefully. Too many though are coming out troubled as I see it. Be glad if you missed out on the fossil fuel pigging, if you were so lucky. You likely have more skills and strength than the men who push hydraulic control levers and get things done around here. Inhale those diesel fumes and you know something is wrong. How hidden away could it be? Could this explain the need to control so much? I can't help but feel that something is this deeply wrong with our culture. The level of absurdity that we've come to has to break. Skepticism and gullibility have to be refreshed as the integrity of the system takes a dive so deep.

[sergi o]

agree

[sergi o]

no, it does not. you are stubbornly stupid.

[sergi o]

liar.

And previously most fractions were not indexed.

liar.

>
> Regards, WM

[sergi o]

diversion.

[Fritz Feldhase]

On Monday, May 23, 2022 at 2:26:04 PM UTC+2, WM wrote:
> zelos...@gmail.com schrieb am Montag, 23. Mai 2022 um 07:06:09 UTC+2:
>

> All natnumbers [...] have an infinite distance from omega. This distance is existing but undefinable.

Oh, it is "undefinable" but you are still talking about "it"? (Since "it" is existing - somehow?)

Fascinating - not!

Hau ab, Mückenheim!

[Ross A. Finlayson]

It's all "resources and rates".

Whatever resources there are
make an automatic machine - ....

Numerical resources, ..., reading and writing them.

Concrete mathematics and finite combinatorics, and
here rates in growing terms, yes make for examples
like "these terms evolve in these rates" why the truncating
part for example, assumes constant-time sorting.

I.e. WM's terms left sitting about have their usual meaning.

Which conscientious mathematicians are expected to "know".

[sergi o]

your vagueness is exceeded by lack of resources.

[Jim Burns]

On 5/23/2022 8:20 AM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 22. Mai 2022 um 18:45:38 UTC+2:

>>> For that assumption you have to claim that the
>>> sequence +1, -1, +1, -1, +1, -1, ... will acumulate
>>> infinitely many units. I will not join you. Hardly
>>> will any mathematician unless he is told that
>>> otherwise set theory would be wrong.
>>
>> If,
>> for each BEFORE and AFTER =< m
>
> There is no relevance of BEFORE and AFTER.

For each entry in sequence 1,2,3,4...
for each split BEFORE and AFTER =< that entry,
some j ends BEFORE and j+1 ends AFTER.

That describes each of the entries, rows, and columns
of a matrix in which the first column of X's
can match the whole matrix of X's and O's

X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
... ... ... ...

This is how they match:

1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ...
2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ...
3/1:6/1 3/2:9/1 3/3:13/1 3/4:18/1 ...
4/1:10/1 4/2:14/1 4/3:19/1 4/4:25/1 ...
5/1:15/1 5/2:20/1 5/3:26/1 5/4:33/1 ...
6/1:21/1 6/2:27/1 6/3:34/1 6/4:42/1 ...
7/1:28/1 7/2:35/1 7/3:43/1 7/4:52/1 ...
8/1:36/1 8/2:44/1 8/3:53/1 8/4:63/1 ...
9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ...

... ... ... ...

An entry is m/n:k/1 such that
k = (m+n-1)*(m+n-2)/2+m

For each m and n in 1,2,3,4,...
k is in 1,2,3,4,...

This is a consequence of the description I gave of
being in 1,2,3,4,...

For each entry in sequence 1,2,3,4...
for each split BEFORE and AFTER =< that entry,
some j ends BEFORE and j+1 ends AFTER.

> There is relevance only of the fact that
> every indexed fraction can get an index
> only from a previously indexd fraction.

Each indexed fraction gets its index from
the first column.

> And previously most fractions were not indexed.

Each fraction is indexed.

Not all collections have Bob-conservation.

[Jim Burns]

On 5/23/2022 8:23 AM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 22. Mai 2022 um 20:49:56 UTC+2:
>> On 5/22/2022 9:32 AM, WM wrote:

>>> That means, there is a permutation such that
>>> the X of the first column

X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...

... ... ... ...

>>> by being exchanged with the O's cover all
>>> matrix positions.
>>
>> Yes.
>
> By exchanging X and O the shares will remain constant.

Not all collections have Bob-conservation.

1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ...
2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ...
3/1:6/1 3/2:9/1 3/3:13/1 3/4:18/1 ...
4/1:10/1 4/2:14/1 4/3:19/1 4/4:25/1 ...
5/1:15/1 5/2:20/1 5/3:26/1 5/4:33/1 ...
6/1:21/1 6/2:27/1 6/3:34/1 6/4:42/1 ...
7/1:28/1 7/2:35/1 7/3:43/1 7/4:52/1 ...
8/1:36/1 8/2:44/1 8/3:53/1 8/4:63/1 ...
9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ...

... ... ... ...

[sergi o]

On 5/19/2022 1:32 PM, WM wrote:
> Jim Burns schrieb am Donnerstag, 19. Mai 2022 um 19:23:45 UTC+2:

>> On 5/19/2022 9:44 AM, WM wrote:
>
>>> That does not exclude aleph_0 dark numbers
>>> immediately before omega.

>> Both dark numbers and omega are irrelevant to
>> natural-number induction.
>
> Of course. Induction covers only the collection of definable natural numbers.

> That does not exclude aleph_0 dark numbers immediately before omega.
>

>> You (WM) do not use "actual infinity" in the usual way.
>
> I use it in the correct way: A quantity larger than all finite quantities. "In spite of significant difference between the notions of the potential and actual infinite, where the former is a variable finite magnitude, growing above all limits, the latter a constant quantity fixed in itself but beyond all finite magnitudes, it happens deplorably often that the one is confused with the other." [Cantor, p. 374]
>
> But you succeed always to distract from the clear evidence proving dark fractions:
>
> We check the number of indexes by bijecting them with the fractions of the first column (we could use every other column or line as well). When applying the indexes for indexing fractions such that m/n gets the index

> k = (m + n - 1)(m + n - 2)/2 + m

> with the resulting sequence

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

> then the integer fractions have to supply these indexes. They are stripped off these indexes. The number of not indexed fractions remains constant.
>
> All definable fractions get indexed. Most fractions don 't get indexed. Dark fractions.
>
> Regards, WM