On 1/14/2023 12:34 PM, WM wrote:
> Jim Burns schrieb am Samstag,
> 14. Januar 2023 um 18:19:26 UTC+1:
>> On 1/14/2023 6:34 AM, WM wrote:

>>> Once upon a time Jim Burns wrote
>>> a sensible sentence:

>>>>> William schrieb am Freitag,
>>>>> 13. Januar 2023 um 17:56:10 UTC+1:

>>>>>> Indeed and among them
>>>>>> no there is no first term you pass.

>> On 1/14/2023 6:34 AM, WM wrote:

>>> True.
>>> The dark numbers have no discernible order.
>
> But then only nonsense.

What is ℕ?
⋃𝓕𝓲𝓷 ⊆ ℕ ⊆ ⋂𝓘𝓷𝓭

You disagree.
Why do you disagree?

If a sequence of claims has a possibly-false
claim, then it has a first possibly-false claim.

You disagree.
Why do you disagree?

If
claims 𝓅 and 𝓅⇒𝓆 precede claim 𝓆
then
𝓆 is not-first-possibly-false.

Either
one of 𝓅 𝓅⇒𝓆 is earlier-possibly-false
than 𝓆
or
both of 𝓅 𝓅⇒𝓆 are not-possibly-false
and 𝓆 is not-possibly-false.

You disagree.
Why do you disagree?

On Saturday, January 14, 2023 at 6:19:26 PM UTC+1, Jim Burns wrote:

> We are not Chuck Norris.

At least most of us (with one exception).

On 1/14/2023 11:34 AM, WM wrote:
> Jim Burns schrieb am Samstag, 14. Januar 2023 um 18:19:26 UTC+1:
>> On 1/14/2023 6:34 AM, WM wrote:
>>> Once upon a time Jim Burns wrote a sensible sentence:
>
>>>>>> Indeed and among them
>>>>>> no there is no first term you pass.
>>>
>>> True.
>>> The dark numbers have no discernible order.
>
> But then only nonsense.
>
> Regards, WM

you, WM, started this nonsense of dark numbers and have be unable to prove any of it over decades.

Do you accept your defeat ?

On Saturday, January 14, 2023 at 6:21:14 PM UTC+1, WM wrote:

> Going from 0 to 1 and trying to identify the first unit fraction encountered is a real task.

Well, if you say so.

The problem is just that there is no _first_ unit fraction one encounters when going from 0 to 1.

So you will necessarily fail when trying. :-)

You might try to find the first even prime which is > 2 too. (Or the largest natural number. How about the round square?)

On Saturday, January 14, 2023 at 7:13:10 PM UTC+1, Jim Burns wrote:
> On 1/14/2023 12:34 PM, WM wrote:
> >
> > But then only nonsense.
> >
> [...]
>
> You disagree.
> Why do you disagree?
"Das liegt daran, dass die Mengenlehre ein inkohärenter Unsinn ist." (WM, de.sci.mathematik)
["That's because set theory is incoherent nonsense."]

So please "spare your silly proofs which have brain-damaged so many mathematicians." (WM, sci.math)

On Saturday, January 14, 2023 at 7:56:30 PM UTC+1, Fritz Feldhase wrote:
> On Saturday, January 14, 2023 at 6:21:14 PM UTC+1, WM wrote:
> >
> > Going from 0 to 1 and trying to identify the first unit fraction encountered is a real task.
> >
> Well, if you say so.
>
> The problem is just that there is no _first_ unit fraction one encounters when going from 0 to 1.
>
> So you will necessarily fail when trying. :-)
>
> You might try to find the first even prime which is > 2 too. (Or the largest natural number. How about the round square?)

Your "task" is realted to the following one:

=========================================================
_Dichotomy paradox (by Zeno)_

"That which is in locomotion must arrive at the half-way stage before it arrives at the goal."
— as recounted by Aristotle, Physics VI:9, 239b10

Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on.

The resulting sequence can be represented as: { ..., 1/16, 1/8, 1/4, 1/2, 1 }

This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.

This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can be neither completed nor begun, and so all motion must be an illusion.

=========================================================
See: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Dichotomy_paradox

So you are concerned with Zeno's paradoxes by now?

On Saturday, 14 January 2023 at 11:06:02 UTC-4, Sergi o wrote:
> On 1/14/2023 5:25 AM, WM wrote:
> > "Here's how we define the uncomputable diagonal number 0 < r < 1. Consider the kth program in our list. If it is syntactically invalid, or if the kth program never outputs a kth digit, or if the kth digit output by the kth program isn't a 3, pick 3 as the kth digit of r. Otherwise, if the kth digit output by the kth program is a 3, pick 4 as the kth digit of r.[Gregory J. Chaitin: "How real are real numbers?", arXiv (2004)]
> Bogus, its from arXiv, anyone can post anything there un-reviewed.

I wouldn't dismiss Chaitin quite so easily (https://en.wikipedia.org/wiki/Gregory_Chaitin). As a computer scientist, I would wager he is mostly a constructivist --- which is a perfectly normal and acceptable position. The only problem is that WM seems to think that anyone who finds even the slightest problem with ZFC is actually and automatically supporting WM's own bullshit --- that seems to be WM's version of "tertium non datur".

On 1/14/2023 2:40 PM, Gus Gassmann wrote:
> On Saturday, 14 January 2023 at 11:06:02 UTC-4, Sergi o wrote:
>> On 1/14/2023 5:25 AM, WM wrote:
>>> "Here's how we define the uncomputable diagonal number 0 < r < 1. Consider the kth program in our list. If it is syntactically invalid, or if the kth program never outputs a kth digit, or if the kth digit output by the kth program isn't a 3, pick 3 as the kth digit of r. Otherwise, if the kth digit output by the kth program is a 3, pick 4 as the kth digit of r.[Gregory J. Chaitin: "How real are real numbers?", arXiv (2004)]
>> Bogus, its from arXiv, anyone can post anything there un-reviewed.
>
> I wouldn't dismiss Chaitin quite so easily (https://en.wikipedia.org/wiki/Gregory_Chaitin). As a computer scientist, I would wager he is mostly a constructivist --- which is a perfectly normal and acceptable position. The only problem is that WM seems to think that anyone who finds even the slightest problem with ZFC is actually and automatically supporting WM's own bullshit --- that seems to be WM's version of "tertium non datur".

Im sure WM picked Chaitin quote via google search, because Chaitin used the word 'pick'

something I really admire are many of the Professors (Math EE) I had in college, just brilliant, open + honest, would discuss anything with you, It was
an honor taking their classes (they were very hard!) One Prof invented the JPEG format (discrete cosine transform).

On Saturday, January 14, 2023 at 4:06:02 PM UTC+1, Sergi o wrote:
> On 1/14/2023 5:25 AM, WM wrote:
> > William schrieb am Samstag, 14. Januar 2023 um 00:13:11 UTC+1:
> >>
> >> "picked" is a nonsense term.

_in this context_.

> The statement "individually pickable numbers" has no meaning.

Indeed!

> > "Here's how we define the uncomputable diagonal number 0 < r < 1. Consider the kth program in our list. If it is syntactically invalid, or if the kth program never outputs a kth digit, or if the kth digit output by the kth program isn't a 3, pick 3 as the kth digit of r. Otherwise, if the kth digit output by the kth program is a 3, pick 4 as the kth digit of r. [Gregory J. Chaitin: "How real are real numbers?", arXiv (2004)]

"Pick" refers to an "action" here, while WM's "pickable" refers to a "property" of a number. A "propery" which is not actually defined - read: is nonsensical. (Are there numbers which are not "individually pickable"? Who knows?!)

Conclusion: WM is dumb like shit!

On Friday, January 13, 2023 at 7:45:11 PM UTC-8, Gus Gassmann wrote:
> On Friday, 13 January 2023 at 17:34:54 UTC-4, FredJeffries wrote:
> > On Friday, January 13, 2023 at 1:13:47 PM UTC-8, Gus Gassmann wrote:
> > > On Friday, 13 January 2023 at 13:44:57 UTC-4, Sergi o wrote:
> >
> > > > If you arrange the natural numbers in alphabetical order, Zero is the last element.
> > > If you refer to the natural numbers by their digit sequences in English, e.g., "one seven nine" for 179, etc, and arrange these digit sequences alphabetically, you get a pretty funky order on the naturals. (I got this from Joel David Hamkins, https://joeldavidhamkins.substack.com/p/the-book-of-numbers)
> > But if you use, say, German digit names, the order type is very different (see his Questions for further thought # 3)
> >
> > https://joeldavidhamkins.substack.com/i/94304826/questions-for-further-thought
> Point taken. (And thanks, BTW, for suggesting Hamkins in the first place)

And using Spanish will give a third order type.

On Saturday, 14 January 2023 at 15:37:48 UTC+2, Fritz Feldhase wrote:
> On Saturday, January 14, 2023 at 12:28:40 PM UTC+1, Eram semper recta wrote:
>
> That's indeed a fascinating claim:

It's TRUE and not my claim. It just is so. A careful study of Euclid's Elements (the foundation of everything STEM) reveals no use of logic as YOU understand it. The subject of logic today has nothing to do with mathematics. Neither does set theory.

> > In any case, logic is not part of mathematics. There is no logic required in geometry and anything that was ever worthwhile came from geometry.

> Says a lot - about you.

Only this - that I am a genius and that you are an absolute moron.

On Saturday, 14 January 2023 at 13:34:06 UTC+2, WM wrote:
> Jim Burns schrieb am Freitag, 13. Januar 2023 um 23:14:45 UTC+1:
> > On 1/13/2023 12:51 PM, WM wrote:
> > > William schrieb am Freitag,
> > > 13. Januar 2023 um 17:56:10 UTC+1:
> > >> On Friday, January 13, 2023
> > >> at 6:14:57 AM UTC-4, WM wrote:
> > >>> William schrieb am Donnerstag,
> > >>> 12. Januar 2023 um 20:08:46 UTC+1:
> > >>>> On Thursday, January 12, 2023
> > >>>> at 2:23:58 PM UTC-4, WM wrote:
> >
> > >>>>> You must know there is a realm
> > >>>>> without such terms
> > >>>>> and there is a realm where
> > >>>>> many terms have already been passed.
> > >>>>
> > >>>> True
> > >>>>
> > >>>>> In between there must be a border.
> > >>>>
> > >>>> Indeed, this border is a bound
> > >>>> (in the present case 0).
> > >>>> A bound does not imply a last element.
> > >>>
> > >>> But when going from the bound
> > >>> to stop at any term desired of the sequence,
> > >>> you have passed infinitely many terms,
> > >>
> > >> Indeed and among them
> > >> no there is no first term you pass.
> True. The dark nubers have no discernible order.

Indeed so. If x be a dark number, then x+1 will be dark? We have a conundrum here:

1. If we say x+1 is dark, then it suggests that we have discerned (identified) x.
2. If we day that x+1 is not dark, then it contradicts the fact that we have not identified x because it is dark.

Mainstream logic has ZERO to do with mathematics.

> > >> Your choices are to pass none or
> > >> to pass a set of cardinality aleph_0
> > >
> > > That means
> > > you can pass them only collectively
> > > without picking them individually.
> > Yes,
> But you can pick some numbers individually.
>
> That is the difference between dark and visible.
>
> > No dark numbers.
>
> Look, you cannot pick dark numbers. Your "proofs" deny their existence. This is an inconsistency, if actual infinity and with it dark numbers are existing. What do you conclude?
>
> Regards, WM

Eram semper recta schrieb am Sonntag, 15. Januar 2023 um 07:00:19 UTC+1:
> On Saturday, 14 January 2023 at 13:34:06 UTC+2, WM wrote:
> > Jim Burns schrieb am Freitag, 13. Januar 2023 um 23:14:45 UTC+1:
> > > On 1/13/2023 12:51 PM, WM wrote:

> > True. The dark numbers have no discernible order.
> Indeed so. If x be a dark number, then x+1 will be dark? We have a conundrum here:
>
> 1. If we say x+1 is dark, then it suggests that we have discerned (identified) x.

No. If you say 17, then you have discerned a number. If you say x, you have not.
> 2. If we day that x+1 is not dark, then it contradicts the fact that we have not identified x because it is dark.

The proof is more than simple. When going from 0 to 1 you will pass infinitely many undistinguishable unit fractions before you can pick one. You can "prove" in set theore that every one can be picked. This shows that set theory is nonsense.

Regards, WM

Fritz Feldhase schrieb am Samstag, 14. Januar 2023 um 22:53:55 UTC+1:

> > > "Here's how we define the uncomputable diagonal number 0 < r < 1. Consider the kth program in our list. If it is syntactically invalid, or if the kth program never outputs a kth digit, or if the kth digit output by the kth program isn't a 3, pick 3 as the kth digit of r. Otherwise, if the kth digit output by the kth program is a 3, pick 4 as the kth digit of r. [Gregory J. Chaitin: "How real are real numbers?", arXiv (2004)]
> "Pick" refers to an "action" here, while WM's "pickable" refers to a "property" of a number.

This property is defined by an action, or better by the impossibility of an action.

Regards, WM

Fritz Feldhase schrieb am Samstag, 14. Januar 2023 um 19:56:30 UTC+1:
> On Saturday, January 14, 2023 at 6:21:14 PM UTC+1, WM wrote:
>
> > Going from 0 to 1 and trying to identify the first unit fraction encountered is a real task.
> Well, if you say so.
>
> The problem is just that there is no _first_ unit fraction one encounters when going from 0 to 1.

If there are all, then there is a first point (x > 0) where you can find one. Alas infinitely many cannot be discerned and hence cannot be found.

Regards, WM

William schrieb am Samstag, 14. Januar 2023 um 18:51:58 UTC+1:
> On Saturday, January 14, 2023 at 1:21:14 PM UTC-4, WM wrote:
> > William schrieb am Samstag, 14. Januar 2023 um 16:22:10 UTC+1:
> > > On Saturday, January 14, 2023 at 7:25:17 AM UTC-4, WM wrote:
> > >
> > >
> > > > > A set does not change.
> > > > Therefore the Peano numbers are not a set. They all can be picked individiually.
> > > The phrase "They all can be picked individually" is just word salad.
> > It is fact
> Nope, it is just meaningless word salad. A Peano set does not change so a Peano set is not "potentially infinite".

It is fact. Try it.

Regards, WM

Jim Burns schrieb am Samstag, 14. Januar 2023 um 19:13:10 UTC+1:
> On 1/14/2023 12:34 PM, WM wrote:
> > Jim Burns schrieb am Samstag,
> > 14. Januar 2023 um 18:19:26 UTC+1:
> >> On 1/14/2023 6:34 AM, WM wrote:
>
> >>> Once upon a time Jim Burns wrote
> >>> a sensible sentence:
> >>>>> William schrieb am Freitag,
> >>>>> 13. Januar 2023 um 17:56:10 UTC+1:
> >>>>>> Indeed and among them
> >>>>>> no there is no first term you pass.
> >> On 1/14/2023 6:34 AM, WM wrote:
>
> >>> True.
> >>> The dark numbers have no discernible order.
> >
> > But then only nonsense.
> What is ℕ?

It depends whether you mean ℕ_Peano, where every n has a FISON and can be picked, or ℕ_Cantor which includes ℕ_Peano but contains more.

> You disagree.
> Why do you disagree?

Because I can prove that dark numbers are existing.

Regards, WM

On Sunday, January 15, 2023 at 12:38:01 PM UTC+1, WM wrote:

> You can "prove" in set theory that every one can be picked.

Please show us this proof.

On Sunday, January 15, 2023 at 12:44:00 PM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Samstag, 14. Januar 2023 um 19:56:30 UTC+1:
> >
> > The problem is just that there is no _first_ unit fraction one encounters when going from 0 to 1.
> >
> [...] there is a first point (x > 0) where [...].

No, there isn't.

Hint: For each and every x e IR, x > 0, there is an n e IN such that 1/n < x.

On Sunday, January 15, 2023 at 7:45:37 AM UTC-4, WM wrote:
>Try it.

"Pickable" is just one of your many many many ways of saying "can be written down"

Each subset of a Peano set of cardinality 1 has the property that it can be written down (Note this does not mean that a set of cardinality aleph_0 that can be written down).

When I "try it" I find that I either pass no elements, or pass a set of l elements with cardinality aleph_0 Since a Peano set does not change each of these elements is part of the set, i.e. I do not add any element so the set.. I cannot "stop at the first element that can be written down".

On 1/15/2023 5:40 AM, WM wrote:
> Fritz Feldhase schrieb am Samstag, 14. Januar 2023 um 22:53:55 UTC+1:
>
>>>> "Here's how we define the uncomputable diagonal number 0 < r < 1. Consider the kth program in our list. If it is syntactically invalid, or if the kth program never outputs a kth digit, or if the kth digit output by the kth program isn't a 3, pick 3 as the kth digit of r. Otherwise, if the kth digit output by the kth program is a 3, pick 4 as the kth digit of r. [Gregory J. Chaitin: "How real are real numbers?", arXiv (2004)]
>> "Pick" refers to an "action" here, while WM's "pickable" refers to a "property" of a number.
>
> This property is defined by an action, or better by the impossibility of an action.

no it is not. one can pick with out removing.

now define it in terms of MATH, or STFU troll.

>
> Regards, WM

On Sunday, 15 January 2023 at 13:38:01 UTC+2, WM wrote:
> Eram semper recta schrieb am Sonntag, 15. Januar 2023 um 07:00:19 UTC+1:
> > On Saturday, 14 January 2023 at 13:34:06 UTC+2, WM wrote:
> > > Jim Burns schrieb am Freitag, 13. Januar 2023 um 23:14:45 UTC+1:
> > > > On 1/13/2023 12:51 PM, WM wrote:
> > > True. The dark numbers have no discernible order.
> > Indeed so. If x be a dark number, then x+1 will be dark? We have a conundrum here:
> >
> > 1. If we say x+1 is dark, then it suggests that we have discerned (identified) x.
> No. If you say 17, then you have discerned a number. If you say x, you have not.

17 is not dark. Read carefully!

> > 2. If we day that x+1 is not dark, then it contradicts the fact that we have not identified x because it is dark.
> The proof is more than simple. When going from 0 to 1 you will pass infinitely many undistinguishable unit fractions before you can pick one. You can "prove" in set theore that every one can be picked. This shows that set theory is nonsense.
>
> Regards, WM

On 1/15/2023 6:48 AM, WM wrote:
> Jim Burns schrieb am Samstag,
> 14. Januar 2023 um 19:13:10 UTC+1:

>> What is ℕ?
>
> It depends whether you mean ℕ_Peano,
> where every n has a FISON and can be picked,
> or ℕ_Cantor
? which includes ℕ_Peano but contains more.

I thought that you would agree that
⋃𝓕𝓲𝓷 ⊆ ℕ_Peano ⊆ ℕ_Cantor ⊆ ⋂𝓘𝓷𝓭

I'll say that another way.

| If,
| for each split between 0 and n
| some i is last-before
| and i⁺⁺ is first-after,
| then
| n is in both ℕ_Peano and ℕ_Cantor

| If
| 0 is in 𝑀 and
| for each j in 𝑀, j⁺⁺ is in 𝑀
| then
| no m in ℕ_Peano and no n in ℕ_Cantor
| is _not_ in (is outside of) 𝑀

Again, I thought you would agree with
those descriptions of ℕ_Peano and ℕ_Cantor

>> You disagree.
>> Why do you disagree?
>
> Because I can prove that
> dark numbers are existing.

But you sneer at our "proofs"
Why aren't you sneering at your own "proofs"?

I have looked at your proofs.
Over and over and over,
your essential step is that
you (WM) _assert_ that there are dark numbers.

My guess is that you're imagining that
there is some authority floating around you,
like perfume, left from your time in authority
over students at Hochschule Augsburg.

That's not how any of this works.

| Either a sequence has no possibly-false claim,
| or it has a first possibly-false claim.

By what authority are those claims asserted?
I don't know, maybe by no authority.
Maybe they aren't even asserted at all.
It doesn't matter.

Separately from any notion of authority,
some claims we can see are not first-possibly-false.

For example,
if
claims 𝓅 and 𝓅⇒𝓆 precede claim 𝓆
then
either
one of 𝓅 𝓅⇒𝓆 is earlier-possibly-false
and 𝓆 is not _first_
or
both of 𝓅 𝓅⇒𝓆 are not-possibly-false
and 𝓆 is not _possibly-false_

In any case,
𝓆 is not _first-possibly-false_

If
we see that 𝓅 and 𝓅⇒𝓆 precede 𝓆
then
we see that
𝓆 is not-first-possibly-false.

In some sequences, for each claim,
we can see that it is not-first-possibly-false.

But then,
no claim in that sequence can be possibly-false.

This is by the authority of our own eyes.

On 1/15/2023 5:48 AM, WM wrote:
> Jim Burns schrieb am Samstag, 14. Januar 2023 um 19:13:10 UTC+1:
>> On 1/14/2023 12:34 PM, WM wrote:
>>> Jim Burns schrieb am Samstag,
>>> 14. Januar 2023 um 18:19:26 UTC+1:
>>>> On 1/14/2023 6:34 AM, WM wrote:
>>
>>>>> Once upon a time Jim Burns wrote
>>>>> a sensible sentence:
>>>>>>> William schrieb am Freitag,
>>>>>>> 13. Januar 2023 um 17:56:10 UTC+1:
>>>>>>>> Indeed and among them
>>>>>>>> no there is no first term you pass.
>>>> On 1/14/2023 6:34 AM, WM wrote:
>>
>>>>> True.
>>>>> The dark numbers have no discernible order.
>>>
>>> But then only nonsense.
>> What is ℕ?
>
> It depends whether you mean ℕ_Peano, where every n has a FISON and can be picked,

Picked ? or Plucked ? pick means it remains, plucked means it is removed...

did you pick or pluck it ?

or ℕ_Cantor which includes ℕ_Peano but contains more.
>
>> You disagree.
>> Why do you disagree?
>
> Because I can prove that dark numbers are existing.

go ahead, we are waiting....

so far none of your dark numbers exist.

you have previously renounced all proofs, did you change your mind ?

>
> Regards, WM

On Sunday, January 15, 2023 at 5:43:06 PM UTC+1, Jim Burns wrote:

> n is in both ℕ_Peano and ℕ_Cantor

Really? How are the sets ℕ_Peano and ℕ_Cantor defined, idiot?

> those descriptions of ℕ_Peano and ℕ_Cantor

Descriptions?

There's a standard definition for IN in the context of set theory (due to von Neumann) which satisfies the Peano axioms (with 0 := { } and s(x) := x u {x}). Now which of ℕ_Peano and ℕ_Cantor is IN?

How about 3 different sets of natural numbers, ℕ_Peano, ℕ_Cantor and IN_von_Neumann?

We might even consider IN_Frege, IN_Dedekind and IN_Russell_Whitehead in addition.