On Monday, June 14, 2021 at 8:08:37 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Montag, 14. Juni 2021 um 15:09:58 UTC+2:
> >
> > HENCE INTERSECTION {E(n) | n e IN} = { }.
> >
> > So each and every natural number has gone.

"has gone" is missleading here. INTERSECTION {E(n) | n e IN} simply doesn't contain any natural numbers (since it is empty).

Hint: The natural numbers didn't leave INTERSECTION {E(n) | n e IN} "one by one" such that "finally" INTERSECTION {E(n) | n e IN} is empty. :-)

> but according to
>
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
>
> only one is lost between two endsegments.

Right. So what?

Hint: INTERSECTION {E(n) | n e IN} is not "the final state" or "the outcome" of a "process", where the natural numbers "one by one" "leave a certain set", or some such nonsense.

INTERSECTION is an "operation" which takes one parameter and "delivers" just one set. Operations in set theory (like UNION, INTERSECTION, POWERSET, etc.) don't work "stepwise".

Try to learn some math, dumbo.

On Saturday, June 12, 2021 at 5:18:28 PM UTC+2, WM wrote:

A classic:

> set theorists have no understanding of their theory.

:-)

F.

On Monday, June 14, 2021 at 3:17:21 PM UTC-3, WM wrote:
> William schrieb am Montag, 14. Juni 2021 um 17:54:08 UTC+2:
> > On Monday, June 14, 2021 at 12:22:46 PM UTC-3, WM wrote:
> > > William schrieb am Montag, 14. Juni 2021 um 02:29:55 UTC+2:
> > WH: The set of definable natural numbers [is] infinite
> > WM: Of course, it is ℕ
> > WM: The set of definable natural numbers is a Peano set.
> > > > Thus there are no dark elements in ℕ\ℕ_def
> > > There are undefined elements.
> > However, there are no undefinable elements. For an element to be dark it is not enough that it is undefined, it must be undefinable.
> There is no sharp border between not yet defined and never defined. We only know from mathematical calculating the abundance that more will never be defined than will ever be defined,

So what? A set of definable elements does not contain any undefinable elements.

> > (a definable natural is an element of a Peano set, is the largest element of a FISON, etc.)
> If you don't know the name of a number you don't know the name of its FISON.

So what? The fact that every element (whether you "know its name or not) of a Peano set is the largest element of a FISON is a fundamental property of a Peabi set,

> But we cannot be sure about FISONs. If dark numbers have dark FISONs

Since ℕ does not contain any dark numbers (all elements of ℕ are definable) it is of no interest whatsoever if dark numbers (whatever they are) have dark FISONs (whatever they are).

--
William Hughes

On 6/14/2021 2:08 PM, WM wrote:
> Greg Cunt schrieb
> am Montag, 14. Juni 2021 um 15:09:58 UTC+2:

>> HENCE INTERSECTION {E(n) | n e IN} = { }.
>>
> So each and every natural number has gone.
> But according to mathematics
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
>
> only one is lost between two endsegments.

Inclusion monotony show that any element in the intersection of
all end segment does not begin an end segment which is followed
by more end segments.

Which elements begin end segments which are followed by
more end segments?
All the elements.

All the elements are not in the intersection of all end segments.

On Saturday, June 12, 2021 at 6:21:55 PM UTC+2, WM wrote:

> only one natural number is lost:
>
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k} (*)

if we compare E(n+1) with E(n) (for any natural number n).

It's easy to prove: An e IN: E(n+1) c E(n) & card(E(n) \ E(n+1)) = 1 .

> But if all endsegments are intersected, the result is the empty set:
>
> ∩{E(k) | k ∈ ℕ } = { }

Indeed. :-)

> This is usually explained by the infinite number of intersected endsegments.

Well, this can actually be formulated as a theorem, dumbo:

Let M c IN, M =/= {}. Then ∩{E(k) | k ∈ M } = { } iff M is infinite.

> But the intersection does not depend on the *number* of endsegments, it depends only on their contents.

Look, dumbo. That's not the way to do math: to formulate half baked ideas without any justification (i.e. proof).

Hint: The intersection depends on the intersected sets, dumbo.

William schrieb am Montag, 14. Juni 2021 um 20:37:19 UTC+2:
> On Monday, June 14, 2021 at 3:17:21 PM UTC-3, WM wrote:

> > There is no sharp border between not yet defined and never defined. We only know from mathematical calculating the abundance that more will never be defined than will ever be defined,

> So what? A set of definable elements does not contain any undefinable elements.

Then the name "definable" seems inappropriate.

Are you unable to understand that aleph_0 numbers exist but only finitely many will get defined. Therefore the rest is undefined.

> > > (a definable natural is an element of a Peano set, is the largest element of a FISON, etc.)
> > If you don't know the name of a number you don't know the name of its FISON.
> So what? The fact that every element (whether you "know its name or not) of a Peano set is the largest element of a FISON is a fundamental property of a Peabi set,
> > But we cannot be sure about FISONs. If dark numbers have dark FISONs
> Since ℕ does not contain any dark numbers (all elements of ℕ are definable)

But never defined.

> it is of no interest whatsoever if dark numbers (whatever they are) have dark FISONs (whatever they are).

Then talk about forever undefined natnumbers. Can you understand that they must exist if only finitely many ever will be defined?

Regards, WM

On 6/14/2021 9:44 AM, WM wrote:
> Sergio schrieb am Montag, 14. Juni 2021 um 01:48:55 UTC+2:
>
>
>> You cannot assume that a "not defined" number, will "never become defined".
>
> That is not assumed but proved. The set of not defined numbers will never have cardinality less than aleph_0. The set of defined numbers will never have cardinality aleph_0. These statements are not objected by any mathematician. Therefore always more numbers will be not defined than defined. Hence there exist numbers which will never become defined.

totally wrong,

1. you ignore the simple fact that *you cannot state the not-defined*
*number k*, *will never become defined*. You do not know the entire
future of everything.

2. your logic is deeply flawed,

"a man has 2 legs
A monkey has 2 legs

therfore, all men are monkeys"

>
> Regards, WM
>

On 6/14/2021 10:20 AM, WM wrote:
> FromTheRafters schrieb am Montag, 14. Juni 2021 um 16:49:12 UTC+2:
>> WM laid this down on his screen :
>>> Sergio schrieb am Montag, 14. Juni 2021 um 01:48:55 UTC+2:
>>>
>>>
>>>> You cannot assume that a "not defined" number, will "never become defined".
>>>
>>> That is not assumed but proved. The set of not defined numbers will never
>>> have cardinality less than aleph_0. The set of defined numbers will never
>>> have cardinality aleph_0. These statements are not objected by any
>>> mathematician. Therefore always more numbers will be not defined than
>>> defined. Hence there exist numbers which will never become defined.
>> Only because you have forced a different notion of defined upon your
>> magical (not natural) set.
>
> That is my definition from which I argue and answer questions.
>
>> All natural numbers are already defined.
>
> Not by digits. There are less than 10^100 posibble sequences in the universe let alone on earth.

See? that illustrates the complete failure of your "defined/not
defined" schtick.

*You do not know what numbers are "defined"*, *no one will ever know*,
*there is no common database*.

Fail.

>
> Regards, WM
>

On 6/14/2021 1:08 PM, WM wrote:
> Greg Cunt schrieb am Montag, 14. Juni 2021 um 15:09:58 UTC+2:
>
>> HENCE INTERSECTION {E(n) | n e IN} = { }.
>>
> So each and every natural number has gone. But according to mathematics
>
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
>
> only one is lost between two endsegments.
>
> Regards, WM
>

BFD

On 6/14/2021 2:50 PM, WM wrote:
> William schrieb am Montag, 14. Juni 2021 um 20:37:19 UTC+2:
>> On Monday, June 14, 2021 at 3:17:21 PM UTC-3, WM wrote:

>>> There is no sharp border between not yet defined and never
>>> defined. We only know from mathematical calculating the
>>> abundance that more will never be defined than will ever
>>> be defined,
>>
>> So what? A set of definable elements does not contain any
>> undefinable elements.
>
> Then the name "definable" seems inappropriate.

It's appropriate if definable xor undefinable.

> Are you unable to understand that aleph_0 numbers exist
> but only finitely many will get defined.
> Therefore the rest is undefined.

Being edible does not imply being eaten.
Being potable does not imply being drunk.
Being legible does not imply being read.

Being definable does not imply being defined.

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

It happens that Greg Cunt formulated :
> On Monday, June 14, 2021 at 5:20:35 PM UTC+2, WM wrote:
>
> "There are less than 10^100 possible sequences in the universe let alone on
> earth." (Wolfgang Mückenheim)
>
> Priceless.

Probably from some deep mathematical understanding of a huge plutonium
atom. Okay, maybe I have my cranks mixed up there - no matter.

On Saturday, June 12, 2021 at 8:35:08 PM UTC+2, WM wrote:

> If no element remains, then all must have gone [...]
>
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k} (*)

You seem to have some genuine problems comprehending basic set theory.

Using (*) we may derive the following theorems in set theory:

E(1) = E(0) \ {0}
E(2) = (E(0) \ {0}) \ {1}
E(3) = ((E(0) \ {0}) \ {1}) \ {2}
etc.

Since we can prove that the "iterated" application of "\" operations allows for swapping of the "operands" we may just drop the parentheses and just write:

E(3) = E(0) \ {0} \ {1} \ {2} .

Still, since each and every set theoretic expression (and hence formula) has to be FINITE in length, writing something like

IN \ {0} \ {1} \ {2} \ ... = { }

is "meaningless". The expression "IN \ {0} \ {1} \ {2} \ ..." is (usually) not defined (if considered as finite expression) and CLEARLY not allowed if considered as infinite expression.

How can we circumvent this litte problem?

(a) We might just DEFINE expressions of the "form"

A \ B_0 \ B_1 \ B_2 \ ...

the following way:

A \ B_0 \ B_1 \ B_2 \ ... := A \ UNION {B_i : n e IN}

Then we would get, for example:

IN \ {0} \ {1} \ {2} \ ... = { } . (Assuming that the terms "{0}", "{1}", "{2}" here refer to the first 3 terms in the sequence (n)_n e IN.)

But it seems to me that some debate might arise concerning the choice of this definition. Especially when dealing with crank. Actually, it seems slightly inappropriate in the context of set theory, imho.

Hence I would like to propose the following approach:

(b) Instead of "iterating" the "\" operation we may define a new set theoretic OPERATION "\\" which is defined the following way:

A \\ B := A \ UNION B .

Hence we may write, say

IN \\ {{0}, {7}, {2}}

instead of

((IN \ {0}) \ {7}) \ {2} or IN \ {0}) \ {7} \ {2} .

Actually, we may (meta-)prove that

"A \\ {B_1, ..., B_n}" refers to the same set as "A \ B_1 \ ... \ B_n"

BUT \\ (in contrast to \) allows NATURALLY for the "infinite case".

It's no problem to state, say

IN \\ {{0}, {1}, {2}, ...} = ...

or more precisely

IN \\ {{n} : n e IN} = ...

Here we would get immediately:

IN \\ {{n} : n e IN} = IN \ UNION {{n} : n e IN} = IN \ IN = { } .

On Monday, June 14, 2021 at 3:50:12 PM UTC-3, WM wrote:
> William schrieb am Montag, 14. Juni 2021 um 20:37:19 UTC+2:
> > On Monday, June 14, 2021 at 3:17:21 PM UTC-3, WM wrote:
>
> > > There is no sharp border between not yet defined and never defined. We only know from mathematical calculating the abundance that more will never be defined than will ever be defined,
>
> > So what? A set of definable elements does not contain any undefinable elements.
> Then the name "definable" seems inappropriate.
>
> Are you unable to understand that aleph_0 numbers exist but only finitely many will get defined. Therefore the rest is undefined.

Indeed, however they are definable.

> > > > (a definable natural is an element of a Peano set, is the largest element of a FISON, etc.)
> > > If you don't know the name of a number you don't know the name of its FISON.
> > So what? The fact that every element (whether you "know its name or not) of a Peano set is the largest element of a FISON is a fundamental property of a Peabi set,
> > > But we cannot be sure about FISONs. If dark numbers have dark FISONs
> > Since ℕ does not contain any dark numbers (all elements of ℕ are definable)
> But never defined.