fredag 30 september 2022 kl. 22:36:00 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Mittwoch, 28. September 2022 um 06:56:38 UTC+2:
> > tisdag 27 september 2022 kl. 20:56:31 UTC+2 skrev WM:
>
> > > The connection is simple: By exchanging X and O never an O is deleted from the matrix. Therefore not all fractions are enumerated.
> > >
> > Your "matrix" and the function N->Q+ are not related.
> Why not? Are the natural numbers from outside of the matrix more than from inside, where they are indexing the integer fractions?
>
> Regards, WM
because there is no "switching around" like you fucking do in a function!

zelos...@gmail.com schrieb am Montag, 3. Oktober 2022 um 07:19:06 UTC+2:
> fredag 30 september 2022 kl. 22:36:00 UTC+2 skrev WM:
> > zelos...@gmail.com schrieb am Mittwoch, 28. September 2022 um 06:56:38 UTC+2:
> > > tisdag 27 september 2022 kl. 20:56:31 UTC+2 skrev WM:
> >
> > > > The connection is simple: By exchanging X and O never an O is deleted from the matrix. Therefore not all fractions are enumerated.
> > > >
> > > Your "matrix" and the function N->Q+ are not related.
> > Why not? Are the natural numbers from outside of the matrix more than from inside, where they are indexing the integer fractions?
> >
> because there is no "switching around" like you fucking do in a function!

For every definable index X the "switching around" can be proved:

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

XXO...
OOO...
XOO...
....

XXO...
XOO...
OOO...
....

Therefore you are wrong. There is "switching around" - at least for definable indices.

Regards, WM

Jim Burns schrieb am Sonntag, 2. Oktober 2022 um 18:51:42 UTC+2:
> On 10/2/2022 5:24 AM, WM wrote:
> > Jim Burns schrieb am Samstag,
> > 1. Oktober 2022 um 22:01:45 UTC+2:
> <WM>
> >>>>> O is not deleted
> >>>>> but it may go to a dark place.
> >> Consider the first-after place.
> >
> > There is none.
> Right.
> That's why O cannot go to a dark place.

There is no first place either where an O could leave the matrix.
You must try to understand thepotential infinite and its blurred border to the dark domain of the actual infinity.
>
> If
> O only goes by steps
> and
> O goes to a dark place
> then
> each split has a first-after place
> and
> there is a first (dark or dark-ish) place

No. Think of the blurred border.
>
> If
> O only goes by steps
> and
> O goes to a dark place
> then, somewhere,
> O takes one step from not-dark to dark.

Same wouldbe true for O leaving the matrix.
>
> If
> O can't take one step from not-dark to dark
> then
> O doesn't go to a dark place.

If O can't take one step from matrix to outside then it can't leave the matrix.
Try to understand the infinite!

> > Dark places cannot be identified.
> > That's why they are dark.
> "... does not exist" and
> "... cannot be identified"
> are not the same thing.

No O can disappear, according to you. But all O's disappear from

XOOO...
XOOO...
XOOO...
XOOO...
....

In the end, when all exchanges of X and O have been carried through according to Cantor we have

XXXX...
XXXX...
XXXX...
XXXX...
....

but no O has left the matrix. Where are they?
> There is an exercise in probability that goes
> something like
> | What is the probability that one of the
> | air molecules from Julius Caesar's last breath
> | is in your lungs right now?
>
> IIRC, it is a very high probability.

Interesting.
>
> It is a probability that the molecule described
> (in your lungs, in JC's last breath) *exists*
>
> Whether it can be identified is a different
> question. I've never heard anyone suggest we can.

The O's are not identifiable in the end (after indexing all definable fractions) although for every identifiable step the result is identifiable too.

> > Only by saying "and so on" or "..."
> > dark places can be occupied.
> "and so on" and "..." _are not dark_
> They stand for "You know what I mean here".

You know it up to potential infinity.
>
> ℕ = {0,1,2,3,...}
>
> stands for
>
> Inductive(ℕ) &
> ~∃𝐵 ⫋ ℕ : Inductive(𝐵)

> '...' _never_ means "We just can't know".
> It's much closer to the opposite of that.

Why can't you count down from omega without a big leap?
You believe that you count all natural numbers when writing 1,2,3,... . But that is an error or a delusion. The big leap over the dark domain occurs in both directions.

Regards, WM

måndag 3 oktober 2022 kl. 10:41:43 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Montag, 3. Oktober 2022 um 07:19:06 UTC+2:
> > fredag 30 september 2022 kl. 22:36:00 UTC+2 skrev WM:
> > > zelos...@gmail.com schrieb am Mittwoch, 28. September 2022 um 06:56:38 UTC+2:
> > > > tisdag 27 september 2022 kl. 20:56:31 UTC+2 skrev WM:
> > >
> > > > > The connection is simple: By exchanging X and O never an O is deleted from the matrix. Therefore not all fractions are enumerated.
> > > > >
> > > > Your "matrix" and the function N->Q+ are not related.
> > > Why not? Are the natural numbers from outside of the matrix more than from inside, where they are indexing the integer fractions?
> > >
> > because there is no "switching around" like you fucking do in a function!
> For every definable index X the "switching around" can be proved:
>
> XOO...
> XOO...
> XOO...
> ...
>
> XXO...
> OOO...
> XOO...
> ...
>
> XXO...
> XOO...
> OOO...
> ...
>
> Therefore you are wrong. There is "switching around" - at least for definable indices.
>
> Regards, WM
your "definable" is meaningless!

There is no switching around, the function just IS!

On 10/3/2022 3:55 AM, WM wrote:
> Jim Burns schrieb am Sonntag, 2. Oktober 2022 um 18:51:42 UTC+2:
>> On 10/2/2022 5:24 AM, WM wrote:
>>> Jim Burns schrieb am Samstag,
>>> 1. Oktober 2022 um 22:01:45 UTC+2:
>> <WM>
>>>>>>> O is not deleted
>>>>>>> but it may go to a dark place.
>>>> Consider the first-after place.
>>>
>>> There is none.
>> Right.
>> That's why O cannot go to a dark place.
>
> There is no first place either where an O could leave the matrix.
> You must try to understand the *potential* infinite and its *blurred border* to the *dark domain* of the *actual infinity*.

Too many Vague terms, REJECTED.

>>
>> If
>> O only goes by steps
>> and
>> O goes to a dark place
>> then
>> each split has a first-after place
>> and
>> there is a first (dark or dark-ish) place
>
> No. Think of the blurred border.

Blurred Ants

Blurred Border Ants

>>
>> If
>> O only goes by steps
>> and
>> O goes to a dark place
>> then, somewhere,
>> O takes one step from not-dark to dark.
>
> Same wouldbe true for O leaving the matrix.
>>
>> If
>> O can't take one step from not-dark to dark
>> then
>> O doesn't go to a dark place.
>
> If O can't take one step from matrix to outside then it can't leave the matrix.

O's do not have legs.

> Try to understand the infinite!
>
>>> Dark places cannot be identified.
>>> That's why they are dark.
>> "... does not exist" and
>> "... cannot be identified"
>> are not the same thing.

> ...
>
> but no O has left the matrix. Where are they?

Where did you put them? This is your gig, and you were the last one touching an O...

>
>> There is an exercise in probability that goes
>> something like
>> | What is the probability that one of the
>> | air molecules from Julius Caesar's last breath
>> | is in your lungs right now?
>>
>> IIRC, it is a very high probability.
>
> Interesting.
>>
>> It is a probability that the molecule described
>> (in your lungs, in JC's last breath) *exists*
>>
>> Whether it can be identified is a different
>> question. I've never heard anyone suggest we can.
>
> The O's are not identifiable in the end (after indexing all definable fractions) although for every identifiable step the result is identifiable too.
>
>>> Only by saying "and so on" or "..."
>>> dark places can be occupied.
>> "and so on" and "..." _are not dark_
>> They stand for "You know what I mean here".
>
> You know it up to potential infinity.
>>
>> ℕ = {0,1,2,3,...}
>>
>> stands for
>>
>> Inductive(ℕ) &
>> ~∃𝐵 ⫋ ℕ : Inductive(𝐵)
>
>> '...' _never_ means "We just can't know".
>> It's much closer to the opposite of that.
>
> Why can't you count down from omega without a big leap?
> You believe that you count all natural numbers when writing 1,2,3,... . But that is an error or a delusion. The big leap over the dark domain occurs in both directions.

total bullshit!

>
> Regards, WM

On 10/3/2022 3:41 AM, WM wrote:
> zelos...@gmail.com schrieb am Montag, 3. Oktober 2022 um 07:19:06 UTC+2:
>> fredag 30 september 2022 kl. 22:36:00 UTC+2 skrev WM:
>>> zelos...@gmail.com schrieb am Mittwoch, 28. September 2022 um 06:56:38 UTC+2:
>>>> tisdag 27 september 2022 kl. 20:56:31 UTC+2 skrev WM:
>>>
>>>>> The connection is simple: By *exchanging* X and O never an O is *deleted* from the matrix. Therefore not all fractions are enumerated.
>>>>>
>>>> Your "matrix" and the function N->Q+ are not related.
>>> Why not? Are the natural numbers from outside of the matrix more than from inside, where they are indexing the integer fractions?
>>>
>> because there is no "*switching around*" like you fucking do in a function!
>
> For every *definable* index X the "*switching around*" can be proved:

wrong. "*switching around*" is not a math term. You cannot prove your imagination.

If you want to be take seriously, USE EQUATIONS.
You do not. Fail.

>
> Therefore you are wrong. There is "*switching around*" - at least for *definable* indices.
>
> Regards, WM

*QuaCk*! words

On 10/3/2022 9:33 AM, Sergi o wrote:
> On 10/3/2022 3:55 AM, WM wrote:
>> Jim Burns schrieb am Sonntag,
>> 2. Oktober 2022 um 18:51:42 UTC+2:

>>> If
>>> O can't take one step from not-dark to dark
>>> then
>>> O doesn't go to a dark place.
>>
>> If O can't take one step from matrix to outside
>> then it can't leave the matrix.
>
> O's do not have legs.

Wait! What's that I see?

XΩΩΩΩ...
XΩΩΩΩ...
XΩΩΩΩ...
XΩΩΩΩ...
XΩΩΩΩ...
...

Upon extensive observation, we find that,
when they've finished migrating,
they attach themselves to a rock,
a dark one, of course,
and mature into their legless form.

Jim Burns was thinking very hard :
> On 10/3/2022 9:33 AM, Sergi o wrote:
>> On 10/3/2022 3:55 AM, WM wrote:
>>> Jim Burns schrieb am Sonntag,
>>> 2. Oktober 2022 um 18:51:42 UTC+2:
>
>>>> If
>>>> O can't take one step from not-dark to dark
>>>> then
>>>> O doesn't go to a dark place.
>>>
>>> If O can't take one step from matrix to outside
>>> then it can't leave the matrix.
>>
>> O's do not have legs.
>
> Wait! What's that I see?
>
> XΩΩΩΩ...
> XΩΩΩΩ...
> XΩΩΩΩ...
> XΩΩΩΩ...
> XΩΩΩΩ...
> ..
>
> Upon extensive observation, we find that,
> when they've finished migrating,
> they attach themselves to a rock,
> a dark one, of course,
> and mature into their legless form.

Ohm at last.

On 10/3/2022 4:55 AM, WM wrote:
> Jim Burns schrieb am Sonntag,
> 2. Oktober 2022 um 18:51:42 UTC+2:
>> On 10/2/2022 5:24 AM, WM wrote:

>>> Only by saying "and so on" or "..."
>>> dark places can be occupied.
>>
>> "and so on" and "..." _are not dark_
>> They stand for "You know what I mean here".
>
> You know it up to potential infinity.

Whatever that is.

'...' means
"You know it for each one we're discussing here"

>> ℕ = {0,1,2,3,...}
>>
>> stands for

∀k : k+1 ≠ 0
∀k : (∀j ≠ k : k+1 ≠ j+1)

Inductive(𝐵) ⟺
0 ∈ 𝐵 ∧ ∀k ∈ 𝐵 : k+1 ∈ 𝐵

>> Inductive(ℕ) ∧
>> ~∃𝐵 ⫋ ℕ : Inductive(𝐵)
>
>> '...' _never_ means "We just can't know".
>> It's much closer to the opposite of that.
>
> Why can't you count down from omega without
> a big leap?

I will ask your (WM's) question this way:

(Q1)
Why can you _step_ down into any FISON 𝐹
∀𝐹 ∈ 𝓕 :
∃m ∉ 𝐹 : m-1 ∈ 𝐹

(Q2)
but you can't _step_ down into their union ⋃𝓕 ?
~∃m ∉ ⋃𝓕 : m-1 ∈ ⋃𝓕

(A1)
Where the FISON 𝐹 ends at k, the successors continue,
and, there at m=k+1, is a place to _step_ down into 𝐹
∀𝐹 ∈ 𝓕 :
∃k ∈ 𝐹 : k+1 ∉ 𝐹

(A2)
FISON 𝐹 ends, but
where the successors continue, to k+1=m,
the _counting-order_ continues,
and there is a _different_ FISON 𝐹' = 𝐹⊕⟨k+1⟩
∀k :
(∃𝐹 ∈ 𝓕 : k ∈ 𝐹) ⟹ (∃𝐹' ∈ 𝓕 : k+1 ∈ 𝐹')

The union ⋃𝓕 of FISONs is that which contains
any k for which a FISON 𝐹 exists which k is in.
∀k :
(k ∈ ⋃𝓕) ⟺ (∃𝐹 ∈ 𝓕 : k ∈ 𝐹)

Therefore,
the union ⋃𝓕 of FISONs does not end.
∀k :
(k ∈ ⋃𝓕) ⟹ (k+1 ∈ ⋃𝓕)

and any m from which _stepping_ down is in ⋃𝓕
is already in ⋃𝓕, and thus is not _into_ ⋃𝓕
∀m :
(m-1 ∈ ⋃𝓕) ⟹ (m ∈ ⋃𝓕)

> You believe that you count all natural numbers
> when writing 1,2,3,... .

...when a variable refers to one of what's in
the minimal inductive set
Inductive(ℕ) ∧ ~∃𝐵 ⫋ ℕ : Inductive(𝐵)

Yes, I do, for metaphorical "count"

> But that is an error or a delusion.
> 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

When we discuss a thing that we can _step_ to,
we discuss an element of ℕ

"But, what about things we can't step to?"
That's a different discussion.

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

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?
>
> > Why can't you count down from omega without
> > a big leap?
> I will ask your (WM's) question this way:
>
> (Q1)
> Why can you _step_ down into any FISON 𝐹
> ∀𝐹 ∈ 𝓕 :
> ∃m ∉ 𝐹 : m-1 ∈ 𝐹
>
> (Q2)
> but you can't _step_ down into their union ⋃𝓕 ?
> ~∃m ∉ ⋃𝓕 : m-1 ∈ ⋃𝓕

Of course you can step down in ⋃𝓕. But, long elaboration short result, every point where you can enter from above has ℵo mainly dark numbers above it and finitely many visible numbers below it.

> > 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. But then we cross infintely many numbers. We cannot count all natural numbers to reach omega.
>
> When we discuss a thing that we can _step_ to,
> we discuss an element of ℕ

No. If omega exists, then we can step to and from omega. Set theorists even have the silly habit to call the well-ordered set of all natnumbers omega.

Regards, WM

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.

<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

Your slightly-later take,
in which leaps are also steps,
is you flailing.

> 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.

No natural number is "all" natural numbers,
the (non-existent) one-to-count-to-for-all,
because, if you can count to k,
then you can count to k+1.

>> When we discuss a thing that we can _step_ to,
>> we discuss an element of ℕ
>
> No.

When we discuss a thing that we can _step_ to,
| meaning,
| for each split, there is
| a last-before i, a first-after j, and j = i+1,
we discuss an element of ℕ

ω is not one of those things.

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!

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.

reversing this process of counting;

I put a rock down, un-count it, then pick it up again.

I can see how this would work on specific natural numbers, like 17...

>>
>>> Why can't you count down from omega without
>>> a big leap?
>> I will ask your (WM's) question this way:
>>
>> (Q1)
>> Why can you _step_ down into any FISON 𝐹
>> ∀𝐹 ∈ 𝓕 :
>> ∃m ∉ 𝐹 : m-1 ∈ 𝐹
>>
>> (Q2)
>> but you can't _step_ down into their union ⋃𝓕 ?
>> ~∃m ∉ ⋃𝓕 : m-1 ∈ ⋃𝓕
>
> Of course you can step down in ⋃𝓕. But, long elaboration short result, every point where you can enter from above has ℵo mainly dark numbers above it and finitely many visible numbers below it.
>
>>> 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. But then we cross infintely many numbers. We cannot count all natural numbers to reach omega.

...and ? ...

>>
>> When we discuss a thing that we can _step_ to,
>> we discuss an element of ℕ
>
> No. If omega exists, then we can step to and from omega.

you cannot step on Omega, unless you draw the symbol on the sidewalk with chalk.

> Set theorists even have the silly habit to call the well-ordered set of all natnumbers omega.

nope, try again...

>
> Regards, WM

On 10/4/2022 3:45 PM, 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!
>
>

one must have the right shoe size for tiny steps too,
else after a while, you would step on billions of these tiny steps all at once...
that could be a skip or a hop,
gasoline powered Pogo stick (Ants) needed

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?

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

On 10/4/2022 2:32 PM, Sergi o wrote:
> On 10/4/2022 3:45 PM, 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!
>>
>>
>
> one must have the right shoe size for tiny steps too,
> else after a while, you would step on billions of these tiny steps all
> at once...
> that could be a skip or a hop,
> gasoline powered Pogo stick (Ants) needed
>

A tiny step. Humm... How about something like:

i[n] = 1/10^n

?

i[0] = 1/10^0 = 1
i[1] = 1/10^1 = .1
i[2] = 1/10^2 = .01
i[3] = 1/10^3 = .001
...

That's pretty small... ;^)

On 10/4/2022 4:45 PM, 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.

I think we need to say what a step is.
In the integers, 4 to 4 is one step.
4 to 2 is two steps == not one step.

I think that the existence of steps
requires the existence of granularity.

> 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!

Consider stepping stones set to help cross
a small stream.

Here, there is one stone in the middle.
Two steps: left bank to stone, stone to
right bank, and you're across.

Over there, for the shorter-legged,
there are three stones. Four steps, and
you're across.

Not a contradiction, different crossings.

But, if there is no granularity, how can
there be steps? A garden snail slooowly
crosses a foot-bridge, just a little further on.
I would say it (eventually) crossed taking
_no_ steps, NOT _infinitely-many_ steps.

There is no granularity in the rationals.
There are no points without between-points.
I would say there are no steps in the
rationals.

Likewise,
I would say there are no steps down from ω
There are no points downward from ω
without points-between.

No points without points-between
is because any step upward toward ω
is not upward to ω,
which is because, if we can count to k,
then we can count to k+1.

On 10/4/2022 3:25 PM, Jim Burns wrote:
> On 10/4/2022 4:45 PM, 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.
>
> I think we need to say what a step is.
> In the integers, 4 to 4 is one step.
> 4 to 2 is two steps == not one step.
[...]

Can traveling from 4 to 2 be accomplished in one, more "direct", step? aka:

4 - 2 = 2

no?

On 10/4/2022 2:00 PM, WM wrote:
> 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

no, you are using those old "broken" and worn out numbers, they brake at n+1,n+2...

what "function" are you talking about ? continuous or f(x,iy) ?

step functions ?

f(t) = 0 for t < 0
= A for t > 0

unit step function u(t) ?

Laplace Transform of Step Function ? or Z transform, Fourier Transform of the step function ?

Extra Credit problem:

how are numbers different from a function ?

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?

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?

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 natural numbers n and m we may call them /consecutive/.

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/.

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?