On 10/15/2022 10:05 AM, Sergi o wrote:
> On 10/12/2022 4:05 PM, Chris M. Thomasson wrote:
>> On 10/12/2022 1:56 PM, WM wrote:
>>> Chris M. Thomasson schrieb am Dienstag, 11. Oktober 2022 um 21:48:30
>>> UTC+2:
>>>> On 10/11/2022 8:32 AM, WM wrote:
>>>
>>>>> Everything easily accessible in a system is not dark in that system.
>>>> Okay. I am thinking of a number. It is not dark to me, but its dark to
>>>> you, right?
>>>
>>> That depends, but may be possible.
>>
>> The number I am thinking of is able to be constructed.
>
> I used to use BAUDOT 5 (ITA2) and paper tape, got good results, AND it
> was recorded so all could see, they called me crazy, but I knew better.

My number can be constructed step-by-step process, stopping at a certain
iteration count. So, I guess its two numbers. One to stop the iteration,
and another that is the result of the iterations function.

>>>> Is that what you mean?
>>>
>>> That is not so important since the border between visible and dark is
>>> blurred. Important is that for everybody the dark part is and remains
>>> infinite, much larger than the always finite visible part:
>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
>>
>> The border between visible and dark? Okay, for some reason this makes
>> me think of the numbers that escape vs the ones that do not in the
>> Mandelbrot set.
>>
>
> yes, those that escape are dark areas, no they are points actually.
>

There are very interesting things one can do with non-escaping points in
the Mandelbrot set. The non-escaping region is saturated with
interesting data to experiment with. An example:

https://fractalforums.org/gallery/1612-300722052941.jpeg

Jim Burns schrieb am Samstag, 15. Oktober 2022 um 20:07:33 UTC+2:
> ⋃𝓕 ⊆ ω and ω ⊆ ⋃𝓕
>
No. ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> ----
> (1)
> ⋃𝓕 ⊆ ω
Of course.
> (2)
> ω ⊆ ⋃𝓕
>
Wrong.
ω = ⋃𝓕 + ∩(infinite Endsegments)
Endsegments are inclusion-monotonic.
∩(inclusion-monotonic infinite sets) is infinite set.
Regards, WM

On 10/16/2022 12:16 PM, WM wrote:
> Jim Burns schrieb am Samstag, 15. Oktober 2022 um 20:07:33 UTC+2:
>
>> ⋃𝓕 ⊆ ω and ω ⊆ ⋃𝓕
>>
> No. ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

that does not even apply!

>> ----
>> (1)
>> ⋃𝓕 ⊆ ω
>
> Of course.
>
>> (2)
>> ω ⊆ ⋃𝓕
>>
> Wrong.
>
> ω = ⋃𝓕 + ∩(infinite Endsegments)

and ∩(all infinite Endsegments) = {}

> Endsegments are inclusion-monotonic.

diversion.

> ∩(inclusion-monotonic infinite sets) is infinite set.

no.

>
> Regards, WM

On 10/16/2022 1:16 PM, WM wrote:
> Jim Burns schrieb am Samstag,
> 15. Oktober 2022 um 20:07:33 UTC+2:

>> (1)
>> ⋃𝓕 ⊆ ω
>
> Of course.
>
>> (2)
>> ω ⊆ ⋃𝓕
>
> Wrong.
>
> ω = ⋃𝓕 + ∩(infinite Endsegments)

I'll call that (⋃𝓕)∪𝔻
where 𝔻 = ∩(infinite Endsegments)

I define 𝛚𝑆 to be the intersection of
all infinite initial sub-segments 𝐵⋮₀ ⊆ 𝑆
𝛚𝑆 = ⋂{𝐵 ⊆ 𝑆 : 𝐵⋮₀}

I restrict the use of '𝛚' to sets 𝑆 which have
at least one infinite initial sub-segment 𝐵⋮₀
(often 𝑆 itself is one),
because ⋂∅ is strange and isn't anything that
we need to discuss, anyway.

Like here, for example.
It's very likely that you (WM) intend for
((⋃𝓕)∪𝔻)⋮₀ to contain all its successors j⁺⁺ and 0
However, whether (⋃𝓕)∪𝔻 does or doesn't,
(⋃𝓕)⋮₀ contains all its successors j⁺⁺ and 0

⋃𝓕 ∈ {𝐵 ⊆ (⋃𝓕)∪𝔻 : 𝐵⋮₀} ≠ ∅
and we can consider
𝛚((⋃𝓕)∪𝔻) = ⋂{𝐵 ⊆ (⋃𝓕)∪𝔻 : 𝐵⋮₀}

It doesn't matter at all what's in 𝔻
ω ⊆ ⋃𝓕

----
For each infinite initial segment 𝐴⋮₀
(𝐴⋮₀ contains all its successors j⁺⁺ and 0),
𝛚𝐴 is the least infinite initial sub-segment ⊆ 𝐴

Also,
𝛚𝐴 is the least infinite initial sub-segment ⊆ 𝐵⋮₀
for any infinite initial segment 𝐵⋮₀
whatever 𝐴⋮₀ is whatever 𝐵⋮₀ is.

| ...because
| the intersection 𝐴∩𝐵 of any 𝐴⋮₀ and any 𝐵⋮₀
| is also an infinite initial segment (𝐴∩𝐵)⋮₀
| and
| 𝐴∩𝐵 is subset both 𝐴 and 𝐵
| | 𝛚𝐵 ⊆ (𝐴∩𝐵)⋮₀ ⊆ 𝐵⋮₀
| | 𝛚𝐴 ⊆ 𝛚𝐵⋮₀ ⊆ (𝐴∩𝐵)⋮₀ ⊆ 𝐴⋮₀
| | 𝛚𝐴 ⊆ 𝛚𝐵 and, similarly, 𝛚𝐵 ⊆ 𝛚𝐴
| | 𝛚𝐴 = 𝛚𝐵

𝛚((⋃𝓕)∪𝔻) = 𝛚(⋃𝓕) = ω

ω ⊆ ⋃𝓕

----
ω ⊆ ⋃𝓕 and ⋃𝓕 ⊆ ω

ω = ⋃𝓕

> Endsegments are inclusion-monotonic.
> ∩(inclusion-monotonic infinite sets)
> is infinite set.

Jim Burns schrieb am Sonntag, 16. Oktober 2022 um 22:36:01 UTC+2:
> On 10/16/2022 1:16 PM, WM wrote:
> > Jim Burns schrieb am Samstag,
> > 15. Oktober 2022 um 20:07:33 UTC+2:
> >> (1)
> >> ⋃𝓕 ⊆ ω
> >
> > Of course.
> >
> >> (2)
> >> ω ⊆ ⋃𝓕
> >
> > Wrong.
> >
> > ω = ⋃𝓕 + ∩(infinite Endsegments)
> I'll call that (⋃𝓕)∪𝔻
> where 𝔻 = ∩(infinite Endsegments)

No matter how you call it, the intersection of inclusion-monotonic sets all of which are larger than X is larger than X. Proof: The sets decrease but have not decreased below X.

Regards, WM

On 10/17/2022 3:23 PM, WM wrote:
> Jim Burns schrieb am Sonntag, 16. Oktober 2022 um 22:36:01 UTC+2:
>> On 10/16/2022 1:16 PM, WM wrote:
>>> Jim Burns schrieb am Samstag,
>>> 15. Oktober 2022 um 20:07:33 UTC+2:
>>>> (1)
>>>> ⋃𝓕 ⊆ ω
>>>
>>> Of course.
>>>
>>>> (2)
>>>> ω ⊆ ⋃𝓕
>>>
>>> Wrong.
>>>
>>> ω = ⋃𝓕 + ∩(infinite Endsegments)
>> I'll call that (⋃𝓕)∪𝔻
>> where 𝔻 = ∩(infinite Endsegments)
>
> No matter how you call it, the intersection of inclusion-monotonic sets all of which are larger than X is larger than X. Proof: The sets decrease but have not decreased below X.

Zippy the Math Pinhead strikes again!

>
> Regards, WM

On Monday, October 17, 2022 at 10:23:07 PM UTC+2, WM wrote:

> [...] the intersection of [the terms of an] inclusion-monotonic [set-sequence] all of which are larger than X is larger than X.

Nope. The intersection is larger than _or equal to_ X.

Consider the inclusion-monotonic set-sequence (S_n) defined with S_n = [0, 1 + 1/n] for all n e IN. Let X = [0, 1].

Then An e IN: X c S_n, but INTERSECTION {S_n : n e IN} = X. qed

Why are that bad at math, Mückenheim?

On 10/17/2022 4:23 PM, WM wrote:
> Jim Burns schrieb am Sonntag,
> 16. Oktober 2022 um 22:36:01 UTC+2:
>> On 10/16/2022 1:16 PM, WM wrote:
>>> Jim Burns schrieb am Samstag,
>>> 15. Oktober 2022 um 20:07:33 UTC+2:

>>>> (1)
>>>> ⋃𝓕 ⊆ ω
>>>
>>> Of course.
>>>
>>>> (2)
>>>> ω ⊆ ⋃𝓕
>>>
>>> Wrong.
>>>
>>> ω = ⋃𝓕 + ∩(infinite Endsegments)
>>
>> I'll call that (⋃𝓕)∪𝔻
>> where 𝔻 = ∩(infinite Endsegments)
>
> No matter how you call it,
> the intersection of inclusion-monotonic sets
> all of which are larger than X
> is larger than X.
> Proof:
> The sets decrease
> but have not decreased below X.

𝛚𝑆 ≜ ⋂{𝐵 ⊆ 𝑆 : 𝐵⋮₀}

⋃𝓕 ⊆ (⋃𝓕)∪𝔻
(⋃𝓕)⋮₀

⋃𝓕 ∈ {𝐵 ⊆ (⋃𝓕)∪𝔻 : 𝐵⋮₀}

⋃𝓕 ⊇ ⋂{𝐵 ⊆ (⋃𝓕)∪𝔻 : 𝐵⋮₀}

⋂{𝐵 ⊆ (⋃𝓕)∪𝔻 : 𝐵⋮₀} = 𝛚((⋃𝓕)∪𝔻)

𝛚((⋃𝓕)∪𝔻) ⊆ ⋃𝓕

----
Let 𝐴⋮₀ and 𝐵⋮₀

if 𝑆⋮₀ ⊆ 𝐵⋮₀
then (𝛚𝐵)⋮₀ ⊆ 𝑆⋮₀ ⊆ 𝐵⋮₀

(𝐴∩𝐵)⋮₀ ⊆ 𝐵⋮₀

(𝛚𝐵)⋮₀ ⊆ (𝐴∩𝐵)⋮₀ ⊆ 𝐵⋮₀

Likewise,
if 𝑆⋮₀ ⊆ 𝐴⋮₀
then (𝛚𝐴)⋮₀ ⊆ 𝑆⋮₀ ⊆ 𝐴⋮₀

(𝛚𝐵)⋮₀ ⊆ (𝐴∩𝐵)⋮₀ ⊆ 𝐴⋮₀

(𝛚𝐴)⋮₀ ⊆ (𝛚𝐵)⋮₀ ⊆ (𝐴∩𝐵)⋮₀ ⊆ 𝐴⋮₀

Therefore,
𝛚𝐴 ⊆ 𝛚𝐵
Similarly,
𝛚𝐵 ⊆ 𝛚𝐴

𝛚𝐴 = 𝛚𝐵

Define ω =
𝛚𝐴 = 𝛚𝐵 = 𝛚((⋃𝓕)∪𝔻) = 𝛚(⋃𝓕)

ω ⊆ ⋃𝓕

Fritz Feldhase schrieb am Montag, 17. Oktober 2022 um 23:33:07 UTC+2:
> On Monday, October 17, 2022 at 10:23:07 PM UTC+2, WM wrote:
>
> > [...] the intersection of [the terms of an] inclusion-monotonic [set-sequence] all of which are larger than X is larger than X.
>
> Nope. The intersection is larger than _or equal to_ X.

You mean the limit. Between all definable infinite terms larger than X and the limit X there are almost all terms. They are dark. They reduce the intersection to X. But this part of the sequence cannot be observed.
>
> Consider the inclusion-monotonic set-sequence (S_n) defined with S_n = [0, 1 + 1/n] for all n e IN. Let X = [0, 1].
>
> Then An e IN: X c S_n, but INTERSECTION {S_n : n e IN} = X. qed

All *definable* terms S_n. But all definable terms have less predecessors than successors. But it is not possible that all terms have more successors than predecessors because the successors belong to a completed set. At omega all terms have been surpassed.

Regards, WM

Jim Burns schrieb am Dienstag, 18. Oktober 2022 um 00:23:11 UTC+2:
> On 10/17/2022 4:23 PM, WM wrote:

> > No matter how you call it,
> > the intersection of inclusion-monotonic sets
> > all of which are larger than X
> > is larger than X.

The limit may be X.
> > Proof:
> > The sets decrease
> > but have not decreased below X.

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

> ⋃𝓕 ⊆ (⋃𝓕)∪𝔻

~∃ n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| < ℵo.

Most of you word salad I cannot read.

Regards, WM

On Tuesday, October 18, 2022 at 10:38:09 AM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Montag, 17. Oktober 2022 um 23:33:07 UTC+2:
> > On Monday, October 17, 2022 at 10:23:07 PM UTC+2, WM wrote:
> >
> > > [...] the intersection of [the terms of an] inclusion-monotonic [set-sequence] all of which are larger than X is larger than X.
> >
> > Nope. The intersection is larger than _or equal to_ X.
> >
> You mean the limit.

The limit *is* the intersection in this case [i.e. a monotonic decreasing sequence], you silly crank.

Anyway, this EXAMPLE ***proves*** my claim (at least the "or equal to" part:

> > Consider the inclusion-monotonic set-sequence (S_n) defined with S_n = [0, 1 + 1/n] for all n e IN. Let X = [0, 1].
> >
> > Then An e IN: X c S_n, but INTERSECTION {S_n : n e IN} = X. qed

AND NOW FUCK OFF, YOU BLITHERING IDIOT!

On Tuesday, October 18, 2022 at 10:46:59 AM UTC+2, WM wrote:

> Most of you word salad I cannot read.

This "word salad" is formulated in the usual language of set-theory, you fucking crank!

AND NOW FUCK OFF, YOU BLITHERING IDIOT!

Fritz Feldhase schrieb am Dienstag, 18. Oktober 2022 um 13:34:38 UTC+2:
> On Tuesday, October 18, 2022 at 10:38:09 AM UTC+2, WM wrote:
> > Fritz Feldhase schrieb am Montag, 17. Oktober 2022 um 23:33:07 UTC+2:
> > > On Monday, October 17, 2022 at 10:23:07 PM UTC+2, WM wrote:
> > >
> > > > [...] the intersection of [the terms of an] inclusion-monotonic [set-sequence] all of which are larger than X is larger than X.
> > >
> > > Nope. The intersection is larger than _or equal to_ X.
> > >
> > You mean the limit.
> The limit *is* the intersection in this case

The intersection of the terms all of which have ℵo more points than X has ℵo more points larger than X. The intersection of all terms is the limit.

Regards, WM

On 10/15/2022 2:55 PM, Chris M. Thomasson wrote:
> On 10/15/2022 10:05 AM, Sergi o wrote:
>> On 10/12/2022 4:05 PM, Chris M. Thomasson wrote:
>>> On 10/12/2022 1:56 PM, WM wrote:
>>>> Chris M. Thomasson schrieb am Dienstag, 11. Oktober 2022 um 21:48:30 UTC+2:
>>>>> On 10/11/2022 8:32 AM, WM wrote:
>>>>
>>>>>> Everything easily accessible in a system is not dark in that system.
>>>>> Okay. I am thinking of a number. It is not dark to me, but its dark to
>>>>> you, right?
>>>>
>>>> That depends, but may be possible.
>>>
>>> The number I am thinking of is able to be constructed.
>>
>> I used to use BAUDOT 5 (ITA2) and paper tape, got good results, AND it was recorded so all could see, they called me crazy, but I knew better.
>
> My number can be constructed step-by-step process, stopping at a certain iteration count. So, I guess its two numbers. One to stop the iteration, and
> another that is the result of the iterations function.
>
>
>>>>> Is that what you mean?
>>>>
>>>> That is not so important since the border between visible and dark is blurred. Important is that for everybody the dark part is and remains
>>>> infinite, much larger than the always finite visible part:
>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
>>>
>>> The border between visible and dark? Okay, for some reason this makes me think of the numbers that escape vs the ones that do not in the Mandelbrot set.
>>>
>>
>> yes, those that escape are dark areas, no they are points actually.
>>
>
> There are very interesting things one can do with non-escaping points in the Mandelbrot set. The non-escaping region is saturated with interesting data
> to experiment with. An example:
>
> https://fractalforums.org/gallery/1612-300722052941.jpeg

looks like clouds,

seems like the fine tenticals are very fine threads of actually "connected" points, or two infinity small points that are adjacent ... (which is
partially dependent on the iteration cut off value) (a graphed line in 2D has connected points too)

On 10/18/2022 10:26 AM, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 18. Oktober 2022 um 13:34:38 UTC+2:
>> On Tuesday, October 18, 2022 at 10:38:09 AM UTC+2, WM wrote:
>>> Fritz Feldhase schrieb am Montag, 17. Oktober 2022 um 23:33:07 UTC+2:
>>>> On Monday, October 17, 2022 at 10:23:07 PM UTC+2, WM wrote:
>>>>
>>>>> [...] the intersection of [the terms of an] inclusion-monotonic [set-sequence] all of which are larger than X is larger than X.
>>>>
>>>> Nope. The intersection is larger than _or equal to_ X.
>>>>
>>> You mean the limit.
>> The limit *is* the intersection in this case
>
> The intersection of the terms all of which have ℵo more points than X has ℵo more points larger than X. The intersection of all terms is the limit.

no, intersection and limit are separate things, there is no linkage between the two.

Intersection is a set operator. It finds elements common to all sets.

your sequence of sets may have a limit.

too bad you are having such a tough time understanding the structure of Math, no wonder, you have it all wadded together.

>
> Regards, WM

On 10/18/2022 4:46 AM, WM wrote:
> Jim Burns schrieb am Dienstag,
> 18. Oktober 2022 um 00:23:11 UTC+2:
>> On 10/17/2022 4:23 PM, WM wrote:

>>> No matter how you call it,
>>> the intersection of inclusion-monotonic sets
>>> all of which are larger than X
>>> is larger than X.
>
> The limit may be X.

Let 𝓢 be some collection of sets 𝑆

The intersection ⋂𝓢 of 𝓢 contains
all and only what is in each 𝑆 in 𝓢
∀x : x ∈ ⋂𝓢 ⟺ (∀𝑆 ∈ 𝓢 : x ∈ 𝑆)

⋂𝓢 is the _greatest lower bound_ of 𝓢
That is to say

⋂𝓢 is a lower bound of 𝓢
⋂𝓢 subset each 𝑆 in 𝓢
∀𝑆 ∈ 𝓢 : ⋂𝓢 ⊆ 𝑆

and
no lower bound 𝐿 exists greater than ⋂𝓢
~∃𝐿 : (∀𝑆 ∈ 𝓢 : ⋂𝓢 ⫋ 𝐿 ⊆ 𝑆)

----
Let 𝓢 be some collection of sets 𝑆

If 𝑆 is not a subset of any 𝑆₂ in 𝓢
then the intersection ⋂𝓢 is not 𝑆
(∃𝑆₂ ∈ 𝓢 : 𝑆 ⊈ 𝑆₂) ⟹ ⋂𝓢 ≠ 𝑆

....because, then, 𝑆 would not be
a lower bound of 𝓢 but ⋂𝓢 is.

| Suppose that, for each 𝑆 in 𝓢 there is
| some 𝑆₂ in 𝓢 to which is 𝑆 is not subset
| ∀𝑆 ∈ 𝓢 : (∃𝑆₂ ∈ 𝓢 : 𝑆 ⊈ 𝑆₂)
| | Then, for each 𝑆 in 𝓢
| the intersection ⋂𝓢 is not 𝑆
| ∀𝑆 ∈ 𝓢 : ⋂𝓢 ≠ 𝑆
| | It is still true, though, that,
| for each 𝑆 in 𝓢
| the intersection ⋂𝓢 is subset 𝑆
| ∀𝑆 ∈ 𝓢 : ⋂𝓢 ⊆ 𝑆
| | so, in this case,
| for each 𝑆 in 𝓢
| the intersection ⋂𝓢 is smaller than 𝑆
| ∀𝑆 ∈ 𝓢 : ⋂𝓢 ⫋ 𝑆

if
∀𝑆 ∈ 𝓢 : (∃𝑆₂ ∈ 𝓢 : 𝑆 ⊈ 𝑆₂)
then
∀𝑆 ∈ 𝓢 : ⋂𝓢 ⫋ 𝑆

Also,
because ⋂𝓢 is the intersection of 𝓢
no 𝐿 larger than ⋂𝓢 is subset all 𝓢
~∃𝐿 : (∀𝑆 ∈ 𝓢 : ⋂𝓢 ⫋ 𝐿 ⊆ 𝑆)

----
>>> No matter how you call it,
>>> the intersection of inclusion-monotonic sets
>>> all of which are larger than X
>>> is larger than X.
>
> The limit may be X.

Consider the set 𝓔 of non-∅ end segments 𝐸

For each 𝐸 in 𝓔 there is
some 𝐸₂ in 𝓔 to which is 𝐸 is not subset
∀𝐸 ∈ 𝓔 : (∃𝐸₂ ∈ 𝓔 : 𝐸 ⊈ 𝐸₂)

For example, for k = 𝑚𝑖𝑛 E
𝐸 ⊈ 𝐸\{k}

You might recall
∀k ∈ ℕ : 𝐸(k+1) = 𝐸(k)\{k}

∀k ∈ ℕ : 𝐸(k) ⊈ 𝐸(k+1)

----
>>> No matter how you call it,
>>> the intersection of inclusion-monotonic sets
>>> all of which are larger than X
>>> is larger than X.
>
> The limit may be X.

Because ⋂𝓔 is the intersection of 𝓔
if
∀𝐸 ∈ 𝓔 : (∃𝐸₂ ∈ 𝓔 : 𝐸 ⊈ 𝐸₂)
then
∀𝐸 ∈ 𝓔 : ⋂𝓔 ⫋ 𝐸

Because
∀k ∈ ℕ : 𝐸(k+1) = 𝐸(k)\{k}

∀𝐸 ∈ 𝓔 : (∃𝐸₂ ∈ 𝓔 : 𝐸 ⊈ 𝐸₂)

Therefore,
∀𝐸 ∈ 𝓔 : ⋂𝓔 ⫋ 𝐸

The intersection ⋂𝓔 of non-∅ end segments
is smaller than each non-∅ end segment
because
∀k ∈ ℕ : 𝐸(k+1) = 𝐸(k)\{k}

On 10/18/2022 8:28 AM, Sergi o wrote:
> On 10/15/2022 2:55 PM, Chris M. Thomasson wrote:
>> On 10/15/2022 10:05 AM, Sergi o wrote:
>>> On 10/12/2022 4:05 PM, Chris M. Thomasson wrote:
>>>> On 10/12/2022 1:56 PM, WM wrote:
>>>>> Chris M. Thomasson schrieb am Dienstag, 11. Oktober 2022 um
>>>>> 21:48:30 UTC+2:
>>>>>> On 10/11/2022 8:32 AM, WM wrote:
>>>>>
>>>>>>> Everything easily accessible in a system is not dark in that system.
>>>>>> Okay. I am thinking of a number. It is not dark to me, but its
>>>>>> dark to
>>>>>> you, right?
>>>>>
>>>>> That depends, but may be possible.
>>>>
>>>> The number I am thinking of is able to be constructed.
>>>
>>> I used to use BAUDOT 5 (ITA2) and paper tape, got good results, AND
>>> it was recorded so all could see, they called me crazy, but I knew
>>> better.
>>
>> My number can be constructed step-by-step process, stopping at a
>> certain iteration count. So, I guess its two numbers. One to stop the
>> iteration, and another that is the result of the iterations function.
>>
>>
>>>>>> Is that what you mean?
>>>>>
>>>>> That is not so important since the border between visible and dark
>>>>> is blurred. Important is that for everybody the dark part is and
>>>>> remains infinite, much larger than the always finite visible part:
>>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
>>>>
>>>> The border between visible and dark? Okay, for some reason this
>>>> makes me think of the numbers that escape vs the ones that do not in
>>>> the Mandelbrot set.
>>>>
>>>
>>> yes, those that escape are dark areas, no they are points actually.
>>>
>>
>> There are very interesting things one can do with non-escaping points
>> in the Mandelbrot set. The non-escaping region is saturated with
>> interesting data to experiment with. An example:
>>
>> https://fractalforums.org/gallery/1612-300722052941.jpeg
>
> looks like clouds,

Yeah. It looks like some sort of nebula.

> seems like the fine tenticals are very fine threads of actually
> "connected" points, or two infinity small points that are adjacent ...
> (which is partially dependent on the  iteration cut off value)     (a
> graphed line in 2D has connected points too)

Every point plotted is connected. The cool thing is that I have still
have my code for this older fractal. It is based on a very special
iterated function system. It's somewhat similar to my MultiJulia
experimental IFS:

http://paulbourke.net/fractals/multijulia

Its fun to work with. Paul did a pretty good job at showing my work.

On 10/18/2022 4:46 AM, WM wrote:
> Jim Burns schrieb am Dienstag,
> 18. Oktober 2022 um 00:23:11 UTC+2:
>> On 10/17/2022 4:23 PM, WM wrote:

>>>> (2)
>>>> ω ⊆ ⋃𝓕
>>>
>>> Wrong.
>>> ω = ⋃𝓕 + ∩(infinite Endsegments)

>> ⋃𝓕 ⊆ (⋃𝓕)∪𝔻
>
> ~∃n ∈ ℕ_def: |ℕ \ {1,2,3,...,n}| < ℵo.
>
> Most of you word salad I cannot read.

For any infinite initial segment,
ω is its least (⊆) infinite initial sub-segment.
(An IIS contains its successors j⁺⁺ and 0)

Denying that that is what ω is
is like denying that 4 comes between 3 and 5
in the natural numbers:
not so much an indication of deeper thinking
as an indication of not knowing what 4 or ω is.

⋃𝓕 contains its successors j⁺⁺ and 0

⋃𝓕 is an infinite initial segment.

Therefore,
ω ⊆ ⋃𝓕

Also ⋃𝓕 ⊆ ω
because
there can be no _first_ j₁ ∈ ⋃𝓕 : j₁ ∉ ω
So, there can be _no_ j ∈ ⋃𝓕 : j ∉ ω
∀j ∈ ⋃𝓕 : j ∈ ω

ω ⊆ ⋃𝓕 and ⋃𝓕 ⊆ ω
ω = ⋃𝓕

----
That's what we mean by ω
the least infinite initial sub-segment.

Proving ω exists is a separate matter.

I define an operator 𝛚
If 𝑆 contains an infinite initial segment,
𝛚𝑆 is the least infinite initial sub-segment of 𝑆

Let 𝐴 and 𝐵 be infinite initial segments.
Then 𝛚𝐴 = 𝛚𝐵

𝐴 and 𝐵 have the same least infinite initial
sub-segment, 𝛚𝐴 = 𝛚𝐵
whatever 𝐴 is, whatever 𝐵 is.

Proof:

𝐴∩𝐵 is also an infinite initial segment.

𝐴∩𝐵 ⊆ 𝐵
𝛚𝐵 is least IIS in 𝐵
𝛚𝐵 ⊆ 𝐴∩𝐵 ⊆ 𝐵

𝐴∩𝐵 ⊆ 𝐴
𝛚𝐵 ⊆ 𝐴∩𝐵 ⊆ 𝐴
𝛚𝐴 is least IIS in 𝐴
𝛚𝐴 ⊆ 𝛚𝐵 ⊆ 𝐴∩𝐵 ⊆ 𝐴

𝛚𝐴 ⊆ 𝛚𝐵
Similarly, 𝛚𝐵 ⊆ 𝛚𝐴
𝛚𝐴 = 𝛚𝐵

We define
ω = 𝛚𝐴 = 𝛚𝐵
the unique least infinite initial sub-segment.

On 10/19/2022 12:46 PM, Jim Burns wrote:
> On 10/18/2022 4:46 AM, WM wrote:
>> Jim Burns schrieb am Dienstag,
>> 18. Oktober 2022 um 00:23:11 UTC+2:
>>> On 10/17/2022 4:23 PM, WM wrote:
>
>>>>> (2)
>>>>> ω ⊆ ⋃𝓕
>>>>
>>>> Wrong.
>>>> ω = ⋃𝓕 + ∩(infinite Endsegments)
>
>>> ⋃𝓕 ⊆ (⋃𝓕)∪𝔻
>>
>> ~∃n ∈ ℕ_def: |ℕ \ {1,2,3,...,n}| < ℵo.
>>
>> Most of you word salad I cannot read.
>
> For any infinite initial segment,
> ω is its least (⊆) infinite initial sub-segment.
> (An IIS contains its successors j⁺⁺ and 0)
>
> Denying that that is what ω is
> is like denying that 4 comes between 3 and 5
> in the natural numbers:
> not so much an indication of deeper thinking
> as an indication of not knowing what 4 or ω is.
>
> ⋃𝓕 contains its successors j⁺⁺ and 0
>
> ⋃𝓕 is an infinite initial segment.
>
> Therefore,
> ω ⊆ ⋃𝓕
>
> Also ⋃𝓕 ⊆ ω
> because
> there can be no _first_ j₁ ∈ ⋃𝓕 : j₁ ∉ ω
> So, there can be _no_ j ∈ ⋃𝓕 : j ∉ ω
> ∀j ∈ ⋃𝓕 : j ∈ ω
>
> ω ⊆ ⋃𝓕 and ⋃𝓕 ⊆ ω
> ω = ⋃𝓕
>
> ----
> That's what we mean by ω
> the least infinite initial sub-segment.
>
> Proving ω exists is a separate matter.
>
> I define an operator 𝛚
> If 𝑆 contains an infinite initial segment,
> 𝛚𝑆 is the least infinite initial sub-segment of 𝑆
>
> Let 𝐴 and 𝐵 be infinite initial segments.
> Then 𝛚𝐴 = 𝛚𝐵
>
> 𝐴 and 𝐵 have the same least infinite initial
> sub-segment, 𝛚𝐴 = 𝛚𝐵
> whatever 𝐴 is, whatever 𝐵 is.
>
> Proof:
>
> 𝐴∩𝐵 is also an infinite initial segment.
>
> 𝐴∩𝐵 ⊆ 𝐵
> 𝛚𝐵 is least IIS in 𝐵
> 𝛚𝐵 ⊆ 𝐴∩𝐵 ⊆ 𝐵
>
> 𝐴∩𝐵 ⊆ 𝐴
> 𝛚𝐵 ⊆ 𝐴∩𝐵 ⊆ 𝐴
> 𝛚𝐴 is least IIS in 𝐴
> 𝛚𝐴 ⊆ 𝛚𝐵 ⊆ 𝐴∩𝐵 ⊆ 𝐴
>
> 𝛚𝐴 ⊆ 𝛚𝐵
> Similarly, 𝛚𝐵 ⊆ 𝛚𝐴
> 𝛚𝐴 = 𝛚𝐵
>
> We define
> ω = 𝛚𝐴 = 𝛚𝐵
> the unique least infinite initial sub-segment.
>
>

The first Endsegment, E(1) is also the infinite initial segment and = ⋃𝓕 = ⋃E

Jim Burns schrieb am Dienstag, 18. Oktober 2022 um 20:16:40 UTC+2:

> The intersection ⋂𝓔 of non-∅ end segments
> is smaller than each non-∅ end segment
> because
> ∀k ∈ ℕ : 𝐸(k+1) = 𝐸(k)\{k}

This shows that the empty intersection cannot be achieved by infinite sets.
All definable k's have ℵo successors. If those were definable too, then we would end with definable finite endsegments.

Note: The intersection of inclusion-monotonic sets cannot be less than the least set. Why should it? Please answer.

If the least set is an ℵo-set, then the intersection is an ℵo-set too.

Regards, WM

WM has brought this to us :
> Jim Burns schrieb am Dienstag, 18. Oktober 2022 um 20:16:40 UTC+2:
>
>> The intersection ⋂? of non-∅ end segments
>> is smaller than each non-∅ end segment
>> because
>> ∀k ∈ ℕ : ?(k+1) = ?(k)\{k}
>
> This shows that the empty intersection cannot be achieved by infinite sets.

It shows no such thing.

> All definable

[...]

> Note: The intersection of inclusion-monotonic sets cannot be less than the
> least set. Why should it? Please answer.

Well, isn't the least set the emptyset?

Jim Burns schrieb am Mittwoch, 19. Oktober 2022 um 19:46:26 UTC+2:
> On 10/18/2022 4:46 AM, WM wrote:

> > ~∃n ∈ ℕ_def: |ℕ \ {1,2,3,...,n}| < ℵo.
> >
> For any infinite initial segment,
> ω is its least (⊆) infinite initial sub-segment.

We had already agreement that there is only one infinite initial segment ℕ.
>
>
> ⋃𝓕 is an infinite initial segment.

No. The union of all FISONs is not larger than all FISONs. All FISONs are finite because only finitely many FISONs can be distinguished by finitely many symbols:

o
oo
ooo
....

Further ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo. These ℵo cannot cannot disappear by forming the union.

Regards, WM

FromTheRafters schrieb am Donnerstag, 20. Oktober 2022 um 13:49:40 UTC+2:
> WM has brought this to us :
> > Jim Burns schrieb am Dienstag, 18. Oktober 2022 um 20:16:40 UTC+2:
> >
> >> The intersection ⋂? of non-∅ end segments
> >> is smaller than each non-∅ end segment
> >> because
> >> ∀k ∈ ℕ : ?(k+1) = ?(k)\{k}
> >
> > This shows that the empty intersection cannot be achieved by infinite sets.
> It shows no such thing.

Only one element can be lost from endsegment to endsegment.
The intersection is always the smallest endsegment. The empty set has finite predecessors.
>
> > Note: The intersection of inclusion-monotonic sets cannot be less than the
> > least set. Why should it? Please answer.
> Well, isn't the least set the emptyset?

Correct.

Regards, WM

On 10/20/2022 7:01 AM, WM wrote:
> FromTheRafters schrieb am Donnerstag, 20. Oktober 2022 um 13:49:40 UTC+2:
>> WM has brought this to us :
>>> Jim Burns schrieb am Dienstag, 18. Oktober 2022 um 20:16:40 UTC+2:
>>>
>>>> The intersection ⋂? of non-∅ end segments
>>>> is smaller than each non-∅ end segment
>>>> because
>>>> ∀k ∈ ℕ : ?(k+1) = ?(k)\{k}
>>>
>>> This shows that the empty intersection cannot be achieved by infinite sets.
>> It shows no such thing.
>
> Only one element can be lost from endsegment to endsegment.
> The intersection is always the smallest endsegment. The empty set has finite predecessors.

Wrong.

1. the empty set is not an endsegment.

2. all endsegments are the same size.

3. "The empty set has finite predecessors" shows stupidity.

>>
>>> Note: The intersection of inclusion-monotonic sets cannot be less than the
>>> least set. Why should it? Please answer.
>> Well, isn't the least set the emptyset?
>
> Correct.
>
> Regards, WM

WM, 100% troll. and always wrong.

On 10/20/2022 6:57 AM, WM wrote:
> Jim Burns schrieb am Mittwoch, 19. Oktober 2022 um 19:46:26 UTC+2:
>> On 10/18/2022 4:46 AM, WM wrote:
>
>>> ~∃n ∈ ℕ_def: |ℕ \ {1,2,3,...,n}| < ℵo.
>>>
>> For any infinite initial segment,
>> ω is its least (⊆) infinite initial sub-segment.
>
> We had already agreement that there is only one infinite initial segment ℕ.

The infinite initial segment is the first endsegment.

>>
>>
>> ⋃𝓕 is an infinite initial segment.
>

<snip bogus troll rot>

>
> Regards, WM