On 10/4/2022 3:57 PM, Chris M. Thomasson wrote:
> On 10/4/2022 3:49 PM, Fritz Feldhase wrote:
>> On Wednesday, October 5, 2022 at 12:39:33 AM UTC+2, Chris M. Thomasson
>> wrote:
>>> On 10/4/2022 3:35 PM, Fritz Feldhase wrote:
>>>> On Tuesday, October 4, 2022 at 10:45:55 PM UTC+2, Chris M. Thomasson
>>>> wrote:
>>>>> On 10/4/2022 1:24 PM, Jim Burns wrote:
>>>>>> On 10/4/2022 3:15 PM, WM wrote:
>>>>>>> Jim Burns schrieb am Montag,
>>>>>>> 3. Oktober 2022 um 19:17:53 UTC+2:
>>>>>>>> On 10/3/2022 4:55 AM, WM wrote:
>>>>>>
>>>>>>>>> The big leap over the dark domain occurs
>>>>>>>>> in both directions.
>>>>>>>>
>>>>>>>> We can't _step_ down into ℕ and
>>>>>>>> we can't _step_ up out of ℕ
>>>>>>>
>>>>>>> We can step from omega to n.
>>>>>>
>>>>>> s/step/leap/
>>>>>>
>>>>>> We can step down from 4 to 3.
>>>>>> We can't step down from 4 to 2.
>>>>>> 4 > 3 > 2
>>>>>>>
>>>>>>> But then we cross infintely many numbers.
>>>>>>>
>>>>>> But then we do not step.
>>>>> [...]
>>>>>
>>>>> Indeed. Well, it depends on the "granularity" of the steps. We can
>>>>> step
>>>>> from 0 to 1. 0 + 1. However, can can go 0 + 1/2 + 1/2 = 1. Step from 0
>>>>> to 1/2, and from 1/2 to 1. Oh shit, there are infinite steps from 0 to
>>>>> 1. Yikes!
>>>>>
>>>> Ehem ... the step stones (we presuppose) are the numbers in IN. Got
>>>> that?
>>>>
>>> Well, that means there is are no gaps from 0...1, wrt natural numbers,
>>> right?
>>
>> Define /gap/. :-P
>>
>> How about: There's a gap between two natural numbers n, m iff there is
>> a natural number k such that n < k < m or m < k < n. ?
>>
>> If there's no gap between two different natural numbers n and m we may
>> call them /consecutive/.
>
> I see. Lets start with two different natural numbers. So, basically
> there is no gap if the absolute value of the difference between two
> _different_ natural numbers is equal to one?
>
> 7 and 6, abs(6-7) = 1, no gap.
>
>  Fair enough?
>

Fair enough wrt restricting ourselves to the natural numbers only. There
is no gap between 3 and 4 because abs(4-3) = 1.

On Wednesday, October 5, 2022 at 12:57:43 AM UTC+2, Chris M. Thomasson wrote:
> On 10/4/2022 3:49 PM, Fritz Feldhase wrote:
> > On Wednesday, October 5, 2022 at 12:39:33 AM UTC+2, Chris M. Thomasson wrote:
> > >
> > > Well, that means there is are no gaps from 0...1, wrt natural numbers, right?
> > >
> > Define /gap/. :-P
> >
> > How about: There's a gap between two natural numbers n, m iff there is a natural number k such that n < k < m or m < k < n. ?
> >
> > If there's no gap between two different natural numbers n and m we may call them /consecutive/.
> >
> I see. Lets start with two different natural numbers. So, basically
> there is no gap if[f] the absolute value of the difference between two
> [...] natural numbers is equal to one?
>
> 7 and 6, abs(6-7) = 1, no gap.
>
> Fair enough?

Sure - if we define /gap/ [in the present context] this way. :-P

You see, no person which is just interested in (using the numbers for) _counting_ finitely many objects will miss a number in the following sequence:

(0), 1, 2, 3, <etc. ad infinitum>

Or stated in Peano's way:

0, s0, ss0, sss0, <etc. ad infinitum>.

No "gaps". :-P

[ Except in WM's worldview, btw. But that's another story. ]

On 10/4/2022 5:49 PM, Fritz Feldhase wrote:
> On Wednesday, October 5, 2022 at 12:39:33 AM UTC+2, Chris M. Thomasson wrote:
>> On 10/4/2022 3:35 PM, Fritz Feldhase wrote:
>>> On Tuesday, October 4, 2022 at 10:45:55 PM UTC+2, Chris M. Thomasson wrote:
>>>> On 10/4/2022 1:24 PM, Jim Burns wrote:
>>>>> On 10/4/2022 3:15 PM, WM wrote:
>>>>>> Jim Burns schrieb am Montag,
>>>>>> 3. Oktober 2022 um 19:17:53 UTC+2:
>>>>>>> On 10/3/2022 4:55 AM, WM wrote:
>>>>>
>>>>>>>> The big leap over the dark domain occurs
>>>>>>>> in both directions.
>>>>>>>
>>>>>>> We can't _step_ down into ℕ and
>>>>>>> we can't _step_ up out of ℕ
>>>>>>
>>>>>> We can step from omega to n.
>>>>>
>>>>> s/step/leap/
>>>>>
>>>>> We can step down from 4 to 3.
>>>>> We can't step down from 4 to 2.
>>>>> 4 > 3 > 2
>>>>>>
>>>>>> But then we cross infintely many numbers.
>>>>>>
>>>>> But then we do not step.
>>>> [...]
>>>>
>>>> Indeed. Well, it depends on the "granularity" of the steps. We can step
>>>> from 0 to 1. 0 + 1. However, can can go 0 + 1/2 + 1/2 = 1. Step from 0
>>>> to 1/2, and from 1/2 to 1. Oh shit, there are infinite steps from 0 to
>>>> 1. Yikes!
>>>>
>>> Ehem ... the step stones (we presuppose) are the numbers in IN. Got that?
>>>
>> Well, that means there is are no gaps from 0...1, wrt natural numbers,
>> right?
>
> Define /gap/. :-P
>
> How about: There's a gap between two natural numbers n, m iff there is a natural number k such that n < k < m or m < k < n. ?
>
> If there's no gap between two different natural numbers n and m we may call them /consecutive/.

yes, stairways are consecutive steps, except at the boundary conditions, the top and bottom.
now an escalator... continuous steps in the middle but with the same boundary conditions.

but where would be the gaps on stairways ? There are gaps in escalator steps, a lady got her skirt caught in one, I saw it.

Extra Credit problems:
would an escalator for cats have smaller steps ?
how about frogs or earthworms ?

On 10/4/2022 6:15 PM, Fritz Feldhase wrote:
> On Wednesday, October 5, 2022 at 12:57:43 AM UTC+2, Chris M. Thomasson wrote:
>> On 10/4/2022 3:49 PM, Fritz Feldhase wrote:
>>> On Wednesday, October 5, 2022 at 12:39:33 AM UTC+2, Chris M. Thomasson wrote:
>>>>
>>>> Well, that means there is are no gaps from 0...1, wrt natural numbers, right?
>>>>
>>> Define /gap/. :-P
>>>
>>> How about: There's a gap between two natural numbers n, m iff there is a natural number k such that n < k < m or m < k < n. ?
>>>
>>> If there's no gap between two different natural numbers n and m we may call them /consecutive/.
>>>
>> I see. Lets start with two different natural numbers. So, basically
>> there is no gap if[f] the absolute value of the difference between two
>> [...] natural numbers is equal to one?
>>
>> 7 and 6, abs(6-7) = 1, no gap.
>>
>> Fair enough?
>
> Sure - if we define /gap/ [in the present context] this way. :-P
>
> You see, no person which is just interested in (using the numbers for) _counting_ finitely many objects will miss a number in the following sequence:
>
> (0), 1, 2, 3, <etc. ad infinitum>
>
> Or stated in Peano's way:
>
> 0, s0, ss0, sss0, <etc. ad infinitum>.
>
> No "gaps". :-P
>
> [ Except in WM's worldview, btw. But that's another story. ]

I usually try to relate WM's math to counting sheeps, it makes better sense that way.

if the sheeps are going between 2 fields and you are the counter, if the sheeps are not numbered, then no gaps.
however if the are numbered then you can calculate gaps, but someone has to get all the sheeps in line, sequentially numbered and that wont work,
especially if you have 300 of them.

On 10/4/2022 4:53 PM, Chris M. Thomasson wrote:
> On 10/4/2022 2:28 PM, Sergi o wrote:
>> On 10/4/2022 2:15 PM, WM wrote:
>>> Jim Burns schrieb am Montag, 3. Oktober 2022 um 19:17:53 UTC+2:
>>>> On 10/3/2022 4:55 AM, WM wrote:
>>>
>>>>> You know it up to potential infinity.
>>>> Whatever that is.
>>>
>>> It is the collection of numbers 1, 2, 3, ... which allow you to return and to reverse the process of counting. Do you believe that all naturalnubers
>>> allow this reversal?
>>
>> so, I pick up a rock, count it, and put it down again.
>
> You are in a "Not holding a rock", state.
>
>
>>
>> reversing this process of counting;
>>
>> I put a rock down,
>
> How can you put a rock down if you are not currently holding it?

ah, the old starting problem, use a magic starter rock ? What if I use both hands, and can pick up/put down 2 rocks asynchronously? nope that wont
fix it.

But I could not do two at a time as I have to put a number on each rock to make sure I did not count it again. Wait, I could use surplus X and Os from
WMs degenerated Matrix, and just put a sticky on each rock.

back in the day, you could have 2 modems "staring" at each other and no communication, as one of them had to initiate, toggle a line(rs232), but both
did not toggle, so they stared waiting.

>
>
>
>> un-count it, then pick it up again.
>>
>> I can see how this would work on specific natural numbers, like 17...
> [...]
>
>

Jim Burns schrieb am Dienstag, 4. Oktober 2022 um 22:24:13 UTC+2:
> On 10/4/2022 3:15 PM, WM wrote:
> >
> > We can step from omega to n.
> s/step/leap/
>
> We can step down from 4 to 3.
> We can't step down from 4 to 2.
> 4 > 3 > 2
> > But then we cross infintely many numbers.
> But then we do not step.

Call it as you like. We can start from omega and end at 17. That is a leap.
>
> <WM>
> >>> Why can't you count down from omega without
> >>> a big leap?
> I still think you were asking
> why _there is no step_ from ω into ℕ

No, I ask why must be infinitely many natnumbers between omega and the end of the leap.
>
> > We cannot count all natural numbers
> > to reach omega.
> Each natural number can be counted to.
> It is what makes a natural number
> a natural number.

But the leap from omega cannot pass less than aleph_0 natnumbers. Why that?
>
> we discuss an element of ℕ
> ω is not one of those things.

The leap from omega to an element of ℕ passes always aleph_0 natnumbers. These cannot be addressed (be the aim of the leap) - neither from above (omega) nor counted to from below (0).

Regards, WM

tisdag 4 oktober 2022 kl. 21:00:51 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Montag, 3. Oktober 2022 um 11:45:08 UTC+2:
>
> > your "definable" is meaningless!
> >
> > There is no switching around, the function just IS!
> Definable number n means: Up to that number n the function can be analyzed in steps.
> Beyond it cannot be analyzed in steps.
>
> Regards, WM

Every natural numbers is reached in finite "steps" so IT APPLIES TO ALL FUCKING NATURAL NUMBERS!

On 10/5/2022 4:10 AM, WM wrote:
> Jim Burns schrieb am Dienstag, 4. Oktober 2022 um 22:24:13 UTC+2:
>> On 10/4/2022 3:15 PM, WM wrote:
>>>
>>> We can step from omega to n.
>> s/step/leap/
>>
>> We can step down from 4 to 3.
>> We can't step down from 4 to 2.
>> 4 > 3 > 2
>>> But then we cross infintely many numbers.
>> But then we do not step.
>
> Call it as you like. We can start from omega and end at 17. That is a leap.

no, that is an endsegment(17)

>>
>> <WM>
>>>>> Why can't you count down from omega without
>>>>> a big leap?
>> I still think you were asking
>> why _there is no step_ from ω into ℕ
>
> No, I ask why must be infinitely many natnumbers between omega and the end of the leap.

goggle "Endsegment" they are always infinite

>>
>>> We cannot count all natural numbers
>>> to reach omega.
>> Each natural number can be counted to.
>> It is what makes a natural number
>> a natural number.
>
> But the leap from omega cannot pass less than aleph_0 natnumbers. Why that?

goggle "Endsegment" they are always infinite, they are born that way.

>>
>> we discuss an element of ℕ
>> ω is not one of those things.
>
> The leap from omega to an element of ℕ passes always aleph_0 natnumbers.
wrong as usual, Deceiver.

goggle "Endsegment" they are always infinite, they are born that way.

>
> Regards, WM

On 10/5/2022 9:06 AM, Sergi o wrote:
> On 10/5/2022 4:10 AM, WM wrote:
>> Jim Burns schrieb am Dienstag, 4. Oktober 2022 um 22:24:13 UTC+2:
>>> On 10/4/2022 3:15 PM, WM wrote:
>>>>
>>>> We can step from omega to n.
>>> s/step/leap/
>>>
>>> We can step down from 4 to 3.
>>> We can't step down from 4 to 2.
>>> 4 > 3 > 2
>>>> But then we cross infintely many numbers.
>>> But then we do not step.
>>
>> Call it as you like. We can start from omega and end at 17. That is a leap.
>
> no, that is an endsegment(17)

I now define Leap(17) as a backwards Endsegment(17)

>>
>> Regards, WM
>
>

On Wednesday, October 5, 2022 at 11:10:59 AM UTC+2, WM wrote:

> We can start [our leap at] omega and [jump to] 17. That is a leap.

So what? You might jump out of the window, too.

> why must [there] be infinitely many natnumbers between omega and the end of the leap.

Because the end of the leap is a natural number, and there are infinitely many natural numbers (finite ordinals) between any natural number and omega. That's why.

> But the leap from omega cannot pass less than aleph_0 natnumbers. Why that?

Because for each and every natural number n there are infinitely many natural numbers (finite ordinals) between n and omega. That's why.

Hope this helps.

On 10/5/2022 11:36 AM, Fritz Feldhase wrote:
> On Wednesday, October 5, 2022 at 11:10:59 AM UTC+2, WM wrote:
>
>> We can start [our leap at] omega and [jump to] 17. That is a leap.
>
> So what? You might jump out of the window, too.
>
>> why must [there] be infinitely many natnumbers between omega and the end of the leap.
>
> Because the end of the leap is a natural number, and there are infinitely many natural numbers (finite ordinals) between any natural number and omega. That's why.
>
>> But the leap from omega cannot pass less than aleph_0 natnumbers. Why that?
>
> Because for each and every natural number n there are infinitely many natural numbers (finite ordinals) between n and omega. That's why.
>
> Hope this helps.

Help him leap into omega

On 10/5/2022 5:10 AM, WM wrote:
> Jim Burns schrieb am Dienstag,
> 4. Oktober 2022 um 22:24:13 UTC+2:
>> On 10/4/2022 3:15 PM, WM wrote:

>>> But then we cross infintely many numbers.
>>
>> But then we do not step.
>
> Call it as you like.
> We can start from omega and end at 17.
> That is a leap.

Where there are only leaps,
I'll call what's NOT there a step.

>> <WM>
>>>>> Why can't you count down from omega without
>>>>> a big leap?
>>
>> I still think you were asking
>> why _there is no step_ from ω into ℕ
>
> No, I ask
> why must be infinitely many natnumbers
> between omega and the end of the leap.

Because _there is no step_ from ω into ℕ

No steps leaves only leaps.
Leaps with steps across splits are finite.
The other leaps are infinite.

----
Why is there no step down into ℕ ?
~∃m ∉ ℕ : m-1 ∈ ℕ

Because there is no step up out of ℕ
~∃k ∈ ℕ : k+1 ∉ ℕ

----
Why is there no step up out of ℕ ?
~∃k ∈ ℕ : k+1 ∉ ℕ

Because ℕ = ⋃𝓕 the union of FISONs
and,
if
n is in ⋃𝓕
then
n is in a FISON 𝐹
and
n+1 is in a FISON 𝐹' = 𝐹⊕⟨n+1⟩
and
n+1 is in ⋃𝓕

On 10/5/2022 7:06 AM, Sergi o wrote:
> On 10/5/2022 4:10 AM, WM wrote:
>> Jim Burns schrieb am Dienstag, 4. Oktober 2022 um 22:24:13 UTC+2:
>>> On 10/4/2022 3:15 PM, WM wrote:
>>>>
>>>> We can step from omega to n.
>>> s/step/leap/
>>>
>>> We can step down from 4 to 3.
>>> We can't step down from 4 to 2.
>>> 4 > 3 > 2
>>>> But then we cross infintely many numbers.
>>> But then we do not step.
>>
>> Call it as you like. We can start from omega and end at 17. That is a
>> leap.
>
> no, that is an endsegment(17)
>
>>>
>>> <WM>
>>>>>> Why can't you count down from omega without
>>>>>> a big leap?
>>> I still think you were asking
>>> why _there is no step_ from ω into ℕ
>>
>> No, I ask why must be infinitely many natnumbers between omega and the
>> end of the leap.
>
>   goggle "Endsegment"  they are always  infinite
>
>>>
>>>> We cannot count all natural numbers
>>>> to reach omega.
>>> Each natural number can be counted to.
>>> It is what makes a natural number
>>> a natural number.
>>
>> But the leap from omega cannot pass less than aleph_0 natnumbers. Why
>> that?
>
>  goggle "Endsegment"  they are always  infinite, they are born that way.
[...]

https://youtu.be/wV1FrqwZyKw

lol. ;^)

>

Fritz Feldhase schrieb am Mittwoch, 5. Oktober 2022 um 18:36:21 UTC+2:
> On Wednesday, October 5, 2022 at 11:10:59 AM UTC+2, WM wrote:
>
> > We can start [our leap at] omega and [jump to] 17. That is a leap.
> > why must [there] be infinitely many natnumbers between omega and the end of the leap.
>
> Because the end of the leap is a natural number, and there are infinitely many natural numbers (finite ordinals) between any natural number and omega. That's why.

If all of them were definable then all of them could be subtracted and nothing below omega would remain.

> > But the leap from omega cannot pass less than aleph_0 natnumbers. Why that?
> Because for each and every natural number n there are infinitely many natural numbers (finite ordinals) between n and omega. That's why.
>
> Hope this helps.

It helps to understand why they are dark. They cannot be removed. They cannot be the ends of leaps. They are undefinable.

Regards, WM

Jim Burns schrieb am Mittwoch, 5. Oktober 2022 um 23:00:29 UTC+2:
> On 10/5/2022 5:10 AM, WM wrote:

> > No, I ask
> > why must be infinitely many natnumbers
> > between omega and the end of the leap.
> Because _there is no step_ from ω into ℕ.

Irrelevant. There is a leap. Why must be infinitely many natnumbers
between omega and the end of every leap into ℕ?

Regards, WM

On 10/6/2022 3:28 PM, WM wrote:
> Jim Burns schrieb am Mittwoch,
> 5. Oktober 2022 um 23:00:29 UTC+2:
>> On 10/5/2022 5:10 AM, WM wrote:

>>> No, I ask
>>> why must be infinitely many natnumbers
>>> between omega and the end of the leap.
>>
>> Because _there is no step_ from ω into ℕ.
>
> Irrelevant. There is a leap.

<WM>
> Why can't you count down from omega without
> a big leap?

Because _there is no step_ from ω into ℕ.

Finite: each split with a (none between) step.
Any split without a (none between) step: infinite.

> Why must be infinitely many natnumbers
> between omega and the end of every leap into ℕ?

There would need to be a none-between leap,
from outside ℕ to inside ℕ
If there are any splits without a step,
then the leap is infinite.

There is nowhere outside ℕ from which
you can step down (none between) into ℕ
~∃m ∉ ℕ : m-1 ∈ ℕ

That's because 𝐹ₘ₋₁⊕⟨m⟩ is a FISON too.

~(m-1 ∈ ℕ ∧ m ∉ ℕ)

Jim Burns schrieb am Donnerstag, 6. Oktober 2022 um 22:35:38 UTC+2:

> There is nowhere outside ℕ from which
> you can step down (none between) into ℕ

Why is that so? Why does every leap cover aleph_0 natural numbers?

> ~∃m ∉ ℕ : m-1 ∈ ℕ

So it is. But there is no gap below omega. A gap could be detected. There is something. The aleph_0 natural numbers covered by every leap. But they are dark, not in a discernible order.

> That's because 𝐹ₘ₋₁⊕⟨m⟩ is a FISON too.

Boring. The collection of FISONs is potentially infinite. All of them have aleph_0 successors. The interesting domain is the natural numbers directly beyond omega. Cantor's claim is that all natural numbers can be counted and used for counting. That is wrong, downwards as well as upwards.
>
> ~(m-1 ∈ ℕ ∧ m ∉ ℕ)

Yes. But below omega there is no gap. No leap can end in a gap. The result of this is the existence of dark nunbers.

Regards, WM

On 10/7/2022 3:09 AM, WM wrote:
> Jim Burns schrieb am Donnerstag,
> 6. Oktober 2022 um 22:35:38 UTC+2:

>> There is nowhere outside ℕ from which
>> you can step down (none between) into ℕ
>
> Why is that so? Why does every leap
> cover aleph_0 natural numbers?

If a leap is _finite_ then,
for each split, a step across it exists.
(last-before and first-after that split exist)
(You _can_ get there from here.)

(Also, of course, there must be
a leap-from place and a leap-to place.)
(A 'there' and a 'here' exist.)

Any leap down from above ℕ into ℕ is _not that_

Leap down from ω ∉ ℕ to n ∈ ℕ
Split that leap {k ≥ n : k ∈ ℕ} {m ≤ ω : m ∉ ℕ}

No i in {k ≥ n : k ∈ ℕ} is last-before
because 𝐹ᵢ⊕⟨i+1⟩ is a FISON too
and i+1 ∈ ℕ too.
No step across {k ≥ n : k ∈ ℕ} {m ≤ ω : m ∉ ℕ}
exists.
Not all splits have steps across.
The leap from ω ∉ ℕ to n ∈ ℕ is _not that_
It is not finite.

>> ~∃m ∉ ℕ : m-1 ∈ ℕ
>
> So it is.
> But there is no gap below omega.
> A gap could be detected.

k ends a FISON 𝐹ₖ
⟺
k is detectable (I guess)
⟺
k is a finite ordinal
⟺
k ∈ ℕ

If ∀k ∈ ℕ : k < m then m is an infinite ordinal.
ω is the first infinite ordinal.
If ∀k ∈ ℕ : k < m then ω ≤ m

If k < ω then k is a finite ordinal.
If k < ω then k is detectable.

Nothing below ω is undetectable.

> There is something.

Not anything undetectable below ω

> There is something.
> The aleph_0 natural numbers covered by
> every leap.

No number is covered by every leap.

Leap from ω ∉ ℕ to n ∈ ℕ

It is not a finite leap, because
the split {k ≥ n : k ∈ ℕ} {m ≤ ω : m ∉ ℕ}
does not have a step across it.

However,
n is not covered by every leap.
n is not covered by the leap from ω to n+1.

> There is something.
> The aleph_0 natural numbers covered by every leap.
> But they are dark, not in a discernible order.
>
>> That's because 𝐹ₘ₋₁⊕⟨m⟩ is a FISON too.
>
> Boring.

A leap from above ℕ into ℕ can't be finite
because
not all splits have (none between) steps
because
𝐹ₘ₋₁⊕⟨m⟩ is a FISON too.

> Cantor's claim is that
> all natural numbers can be counted and
> used for counting.

ℕ ≝ ⋃𝓕

If k ∈ ℕ then k ends FISON 𝐹ₖ

Jim Burns schrieb am Freitag, 7. Oktober 2022 um 20:34:43 UTC+2:
> On 10/7/2022 3:09 AM, WM wrote:
..
> > The aleph_0 natural numbers covered by
> > every leap.
> No number is covered by every leap.

aleph_0 natural numbers are covered by every leap.
aleph_0 numbers > 0 numbers = 0 no number.
> However,
> n is not covered by every leap.
> n is not covered by the leap from ω to n+1.

Right. Definable numbers are not covered. But aleph_0 natural numbers are covered by every leap.

> A leap from above ℕ into ℕ can't be finite
> because
> 𝐹ₘ₋₁⊕⟨m⟩ is a FISON too.

We agree that the leap can end at every desired FISON end.
But aleph_0 natural numbers are covered by every leap.

> If k ∈ ℕ then k ends FISON 𝐹ₖ

then the leap can end at k.
But the leap cannot cover less than aleph_0 numbers.
See the difference?

Regards, WM

On 10/7/2022 12:09 AM, WM wrote:
> Jim Burns schrieb am Donnerstag, 6. Oktober 2022 um 22:35:38 UTC+2:
>
>> There is nowhere outside ℕ from which
>> you can step down (none between) into ℕ
>
> Why is that so? Why does every leap cover aleph_0 natural numbers?
>
>> ~∃m ∉ ℕ : m-1 ∈ ℕ
>
> So it is. But there is no gap below omega. A gap could be detected. There is something. The aleph_0 natural numbers covered by every leap. But they are dark, not in a discernible order.
>
>> That's because 𝐹ₘ₋₁⊕⟨m⟩ is a FISON too.
>
> Boring. The collection of FISONs is potentially infinite. All of them have aleph_0 successors. The interesting domain is the natural numbers directly beyond omega. Cantor's claim is that all natural numbers can be counted and used for counting. That is wrong, downwards as well as upwards.
>>
>> ~(m-1 ∈ ℕ ∧ m ∉ ℕ)
>
> Yes. But below omega there is no gap. No leap can end in a gap. The result of this is the existence of dark nunbers.

With regard to artificially restricting ones self to the natural
numbers, there is no gap between say, [41...42]. However, there is
41+1/2 that can be described by the fractions. Also, since an irrational
can be sort of "described" by a continued fraction, well, is it
interesting that a rational can be used to describe an irrational? Good
ol' 355/113... ;^)

Make any sense? Thanks.

On 10/7/2022 3:42 PM, WM wrote:
> Jim Burns schrieb am Freitag,
> 7. Oktober 2022 um 20:34:43 UTC+2:

>> A leap from above ℕ into ℕ can't be finite
>> because
>> 𝐹ₘ₋₁⊕⟨m⟩ is a FISON too.
>
> We agree that the leap can end at every
> desired FISON end.
> But aleph_0 natural numbers are covered by
> every leap.

For simplicity, consider only 0 and ω
and whatever is between them.

Split them between
what can be reached one-by-one from 0
and what can't be reached that way.

What's needed in order to reach one-by-one?

One-by-one, whichever way they're split
into before and after,
just-one-more must be possible.
For each split,
a from-before-place and a to-after-place,
with none between, must exist.

Each place on the 0-side of the split
can be reached one-by-one from 0

Also,
| the _next place after_ a place which can be
| reached one-by-one can be reached one-by-one.

Therefore,
no just-one-more exists across the
reachable/not-reachable split,
because there is no last-before.
we'd need a last-before and and a first-after,
and there is no last-before.

Therefore,
for any leap from the ω-side to the 0-side
or from the 0-side to the ω-side,
not all splits have just-one-more;
it can't be reached one-by-one.

A leap which can't be reached one-by-one
is _not finite_

>> 𝐹ₘ₋₁⊕⟨m⟩ is a FISON too.

> But aleph_0 natural numbers are covered by
> every leap.

aleph_0 natural numbers are covered by every leap
because
𝐹ₘ₋₁⊕⟨m⟩ is a FISON too
and
the 0-side ω-side split is not just-one-more.

The 0-side ω-side split is not just-one-more,
and also
everything on the 0-side is in a FISON.

That is more often said this way:
there are infinitely-many natural numbers
and also
every natural number is finite.

On 10/4/2022 4:15 PM, Fritz Feldhase wrote:
> On Wednesday, October 5, 2022 at 12:57:43 AM UTC+2, Chris M. Thomasson wrote:
>> On 10/4/2022 3:49 PM, Fritz Feldhase wrote:
>>> On Wednesday, October 5, 2022 at 12:39:33 AM UTC+2, Chris M. Thomasson wrote:
>>>>
>>>> Well, that means there is are no gaps from 0...1, wrt natural numbers, right?
>>>>
>>> Define /gap/. :-P
>>>
>>> How about: There's a gap between two natural numbers n, m iff there is a natural number k such that n < k < m or m < k < n. ?
>>>
>>> If there's no gap between two different natural numbers n and m we may call them /consecutive/.
>>>
>> I see. Lets start with two different natural numbers. So, basically
>> there is no gap if[f] the absolute value of the difference between two
>> [...] natural numbers is equal to one?
>>
>> 7 and 6, abs(6-7) = 1, no gap.
>>
>> Fair enough?
>
> Sure - if we define /gap/ [in the present context] this way. :-P
[...]

Indeed. :^)

One is atomic in the sense that we are artificially restricting
ourselves to the natural numbers. Well, within this strict context, the
number one cannot be broken apart for it is the the smallest unit, so to
speak. ;^)

Chris M. Thomasson schrieb am Freitag, 7. Oktober 2022 um 22:35:00 UTC+2:
> On 10/7/2022 12:09 AM, WM wrote:

> > below omega there is no gap. No leap can end in a gap. The result of this is the existence of dark nunbers.
> With regard to artificially restricting ones self to the natural
> numbers, there is no gap between say, [41...42].

Yes, there is no gap. And if ω exists, then there is no gap between ω and ℕ. But every leap from ω into ℕ covers infinitely many natural numbers.

Reagrds, WM

Jim Burns schrieb am Samstag, 8. Oktober 2022 um 20:39:13 UTC+2:
> On 10/7/2022 3:42 PM, WM wrote:
> > Jim Burns schrieb am Freitag,
> > 7. Oktober 2022 um 20:34:43 UTC+2:
> >> A leap from above ℕ into ℕ can't be finite
> >> because
> >> 𝐹ₘ₋₁⊕⟨m⟩ is a FISON too.
> >
> > We agree that the leap can end at every
> > desired FISON end.
> > But aleph_0 natural numbers are covered by
> > every leap.
> For simplicity, consider only 0 and ω
> and whatever is between them.
>
> Split them between
> what can be reached one-by-one from 0
> and what can't be reached that way.
>
> What's needed in order to reach one-by-one?
>
> One-by-one, whichever way they're split
> into before and after,
> just-one-more must be possible.
> For each split,
> a from-before-place and a to-after-place,
> with none between, must exist.
>
> Each place on the 0-side of the split
> can be reached one-by-one from 0

Every FISON-end can be reached one-by-one. But every FISON contains less than half of the infinite initial segment, i.e., of all natural numbers {1, 2, 3, ...}.
Proof:
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
or
∀n ∈ ℕ_def: 2n < ℵo ==> n < ℵo/2.

Nevertheless it is impossible to distinguish different infinite initial segments. Why? Because the additional numbers are dark.
>
> Also,
> | the _next place after_ a place which can be
> | reached one-by-one can be reached one-by-one.
>
> Therefore,
> no just-one-more exists across the
> reachable/not-reachable split,
> because there is no last-before.
> we'd need a last-before and and a first-after,
> and there is no last-before.

Yes. But you fail to understand potential infinity.
The FISONs are an infinite sequence.
But they never become actually infinite:

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

Cantor however claims the existence of |ℕ| = ℵo.
>
> Therefore,
> for any leap from the ω-side to the 0-side
> or from the 0-side to the ω-side,
> not all splits have just-one-more;
> it can't be reached one-by-one.

Cantor claims that from 0 all natural nunbers can be reached such that none remains unreached. That is precondition for every bijection with ℕ..
>
> A leap which can't be reached one-by-one
> is _not finite_

Correct. It covers dark numbers which cannot be distingusihed. But Cantor claims that from 0 all |ω| natural nunbers can be reached such that none remains unreached.

> > But aleph_0 natural numbers are covered by
> > every leap.
> aleph_0 natural numbers are covered by every leap
> because
> 𝐹ₘ₋₁⊕⟨m⟩ is a FISON too
> and
> the 0-side ω-side split is not just-one-more.
>
>
> The 0-side ω-side split is not just-one-more,
> and also
> everything on the 0-side is in a FISON.

Everything that can be reached one-by-one is in a FISON. But everything that can be reached one-by-one is less than |ω|/2. Don't you believe?
>
> That is more often said this way:
> there are infinitely-many natural numbers

That was often said this way: There are potentially infinitely many natural numbers. But Cantor claimed more: "Wenn zwei wohldefinierte Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für Element, einander zuordnen lassen". Vollständig means complete, |ω|, you know?

> and also
> every natural number is finite.

The infinite set with |ω| elements can never be reached step-by-step. And it can never be completed such that all numbers less than ω are taken. But if they are existing, then they can be taken step-by-step unless they are dark.

Regards, WM

WM has brought this to us :

> Every FISON-end can be reached one-by-one. But every FISON contains less than
> half of the infinite initial segment, i.e., of all natural numbers {1, 2, 3,
> ...}.

A segment of that set cannot be the whole set. The initial segment gets
defined as all elements strictly less than some element, not less than
or equal to some element.

https://proofwiki.org/wiki/Definition:Initial_Segment