Am 24.03.2024 um 21:11 schrieb WM:

> Does ℕ = {1, 2, 3, ...} contain all natural numbers such that none can
> be added?

Well, IN is defined this way. I mean, as the set of _all_ natural numbers.

> If so, then the bijection of ℕ with E = {2, 4, 6, ...} would prove that
> both sets have the same number of elements.

Of course. #IN = aleph_0 = #E.

> Then the completion of E resulting in E* = {1, 2, 3, 4, 5, 6, ...}[.] [Hence the number of E*'s elements] would [be twice] the number of [E's] elements.

Exactly!

Hint: The number of elements in E is aleph_0. And 2 * aleph_0 = aleph_0
+ aleph_0 = aleph_0.

> Then there are more natural numbers than were originally in ℕ.

Nope, since 2 * aleph_0 = aleph_0.

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Am 30.03.2024 um 17:37 schrieb Tom Bola:

> Piss off already, blithering retard...

Grüß Dich. :-)

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Le 30/03/2024 à 19:47, Richard Damon a écrit :
> On 3/30/24 12:01 PM, WM wrote:
>> Le 29/03/2024 à 14:29, Richard Damon a écrit :
>>
>>> And, it can be shown that ℕ and ℚ are the same size.
>>
>> That is believed by some stupids who are too dense to understand logic.
>> The Os in the matrix
>> XOOO...
>> XOOO...
>> XOOO...
>> XOOO...
>> ..
>> cannot disappear by exchange with Xs.
>
> So?
>
> That isn't the Bijection we are looking for.

It is true for every mapping.

Regards, WM

Le 30/03/2024 à 20:04, Jim Burns a écrit :

> Hold back just a moment on assigning labels.
> Is this a fair description of
> what you think we think?
>
> I think that WM thinks that, in our telling,
> one gets to ω
> by going out out out to "infinity"

1, 2, 3, ... ω, ω + 1, ... [Cantor, 1882]

Regards, WM

WM schrieb:

> Le 30/03/2024 à 20:04, Jim Burns a écrit :
>
>> Hold back just a moment on assigning labels.
>> Is this a fair description of
>> what you think we think?
>>
>> I think that WM thinks that, in our telling,
>> one gets to ω
>> by going out out out to "infinity"
>
> 1, 2, 3, ... ω, ω + 1, ... [Cantor, 1882]

This is a sequence of all ordinals including limit ordinals, while your
attempt wants to make statements about the natural numbers 1, 2, 3, ...

On 03/31/2024 06:19 AM, Tom Bola wrote:
> WM schrieb:
>
>> Le 30/03/2024 à 20:04, Jim Burns a écrit :
>>
>>> Hold back just a moment on assigning labels.
>>> Is this a fair description of
>>> what you think we think?
>>>
>>> I think that WM thinks that, in our telling,
>>> one gets to ω
>>> by going out out out to "infinity"
>>
>> 1, 2, 3, ... ω, ω + 1, ... [Cantor, 1882]
>
> This is a sequence of all ordinals including limit ordinals, while your
> attempt wants to make statements about the natural numbers 1, 2, 3, ...
>

Just think of it as a model of modularity,
that modules are of equal size, that modules
have a grain or equi-partitioning, as the
number or count of those increases, and
then as in the infinite limit and continuum limit,
that the boundaries of the modules maintain their
proportion, of course though this is still very "natural".

The modularity of numbers is a key term with trichotomy.

Otherwise this sort of f(n) = n, f(n) = 2n, and so on,
simply reflect as a function and functional relation,
how it so happens that systems of equations include
the algebra of functions, keeping things simple.

Whether natural numbers include infinite numbers,
the infinitely-grand, has that according to naive
expansion of comprehension they do. It's just the
regular ordinary opinion of regular ordinary set
theory, they don't.

There are others, ....

Es gibt anderen, ....

"Sein ein Hengst, nicht ein Zwerg."

Ross Finlayson schrieb:

> On 03/31/2024 06:19 AM, Tom Bola wrote:
>> WM schrieb:
>>
>>> Le 30/03/2024 à 20:04, Jim Burns a écrit :
>>>
>>>> Hold back just a moment on assigning labels.
>>>> Is this a fair description of
>>>> what you think we think?
>>>>
>>>> I think that WM thinks that, in our telling,
>>>> one gets to ω
>>>> by going out out out to "infinity"
>>>
>>> 1, 2, 3, ... ω, ω + 1, ... [Cantor, 1882]
>>
>> This is a sequence of all ordinals including limit ordinals, while your
>> attempt wants to make statements about the natural numbers 1, 2, 3, ...
>>
>
> Just think of it as a model of modularity,
> that modules are of equal size, that modules
> have a grain or equi-partitioning, as the
> number or count of those increases, and
> then as in the infinite limit and continuum limit,
> that the boundaries of the modules maintain their
> proportion, of course though this is still very "natural".
>
> The modularity of numbers is a key term with trichotomy.
>
> Otherwise this sort of f(n) = n, f(n) = 2n, and so on,
> simply reflect as a function and functional relation,
> how it so happens that systems of equations include
> the algebra of functions, keeping things simple.
>
> Whether natural numbers include infinite numbers,
> the infinitely-grand, has that according to naive
> expansion of comprehension they do. It's just the
> regular ordinary opinion of regular ordinary set
> theory, they don't.
>
> There are others, ....
>
> Es gibt anderen, ....
>
>
> "Sein ein Hengst, nicht ein Zwerg."

Where do you see trichotomy? The above sequence lists all existent
limit ordinals (including limit ordinals) winthin a linear order
up to eps_0, does it.

Ross Finlayson schrieb:
> On 03/31/2024 06:19 AM, Tom Bola wrote:
>> WM schrieb:
>>
>>> Le 30/03/2024 à 20:04, Jim Burns a écrit :
>>>
>>>> Hold back just a moment on assigning labels.
>>>> Is this a fair description of
>>>> what you think we think?
>>>>
>>>> I think that WM thinks that, in our telling,
>>>> one gets to ω
>>>> by going out out out to "infinity"
>>>
>>> 1, 2, 3, ... ω, ω + 1, ... [Cantor, 1882]
>>
>> This is a sequence of all ordinals including limit ordinals, while your
>> attempt wants to make statements about the natural numbers 1, 2, 3, ...
>>
>
> Just think of it as a model of modularity,
> that modules are of equal size, that modules
> have a grain or equi-partitioning, as the
> number or count of those increases, and
> then as in the infinite limit and continuum limit,
> that the boundaries of the modules maintain their
> proportion, of course though this is still very "natural".
>
> The modularity of numbers is a key term with trichotomy.
>
> Otherwise this sort of f(n) = n, f(n) = 2n, and so on,
> simply reflect as a function and functional relation,
> how it so happens that systems of equations include
> the algebra of functions, keeping things simple.
>
> Whether natural numbers include infinite numbers,
> the infinitely-grand, has that according to naive
> expansion of comprehension they do. It's just the
> regular ordinary opinion of regular ordinary set
> theory, they don't.
>
> There are others, ....
>
> Es gibt anderen, ....
>
>
> "Sein ein Hengst, nicht ein Zwerg."

Where do you see trichotomy? The above sequence lists all existent
ordinals (including limit ordinals) winthin a linear order up to eps_0,
does it.

Am 30.03.2024 um 17:09 schrieb WM:

> If there are all points on the real line permanently existing, then
> there is a point next to zero.

No, there is no point "next to zero".

If x is a point "close to zero", then the point x/2 is even closer to
zero, Du dummer Spinner.

Geh doch mal zum Psychiater, Mückenheim!

Am 29.03.2024 um 17:31 schrieb Tom Bola:

> Sets are not ordered, so <bla>

Check: https://en.wikipedia.org/wiki/Partially_ordered_set
and: https://mathworld.wolfram.com/OrderedSet.html

Moebius schrieb:

> Am 29.03.2024 um 17:31 schrieb Tom Bola:
>
>> Sets are not ordered, so <bla>
>
> Check: https://en.wikipedia.org/wiki/Partially_ordered_set
> and: https://mathworld.wolfram.com/OrderedSet.html

By default, sets are not ordered.

On 3/31/2024 8:20 AM, WM wrote:
> Le 30/03/2024 à 20:04, Jim Burns a écrit :

>> Hold back just a moment on assigning labels.
>> Is this a fair description of
>> what you think we think?
>>
>> I think that WM thinks that, in our telling,
>> one gets to ω
>> by going out out out to "infinity"
>
> 1, 2, 3, ... ω, ω + 1, ... [Cantor, 1882]

| For ordinals φ with not.finite ⟦0,φ⦆
| the first is ω
| not.WM, 2024

| [Finite set] S can be given a total ordering which
| is well-ordered both forwards and backwards.
| That is, every non-empty subset of S has both
| a least and a greatest element in the subset.
| Paul Stäckel (1862-1919)
https://en.wikipedia.org/wiki/Finite_set

No totally.ordered ⟦0,φ⦆ ⟦0,φ+1⦆ exist such that
every non-empty subset of ⟦0,φ⦆ has both
a least and a greatest element in the subset
and
NOT every non-empty subset of ⟦0,φ+1⦆ has both
a least and a greatest element in the subset.

....because,
for each non.empty Sᵩ₊₁ ⊆ ⟦0,φ+1⦆
either
Sᵩ₊₁ = {φ+1} which has a least and a greatest
or
non.empty Sᵩ₊₁\{φ+1} ⊆ ⟦0,φ⦆
Sᵩ₊₁\{φ+1} has a least and a greatest
Sᵩ₊₁ has a least and a greatest

No ordinal ψ exists such that ψ+1 = ω

....because,
if ψ exists, ψ is finite or infinite.
If ψ is finite, then ω = ψ+1 is finite,
but ω is not finite.
If ψ is infinite, then ω > ψ is not first.infinite.
but ω is first.infinite.

ψ is not finite and not infinite.

ψ not.exists such that ψ+1 = ω
Visibleᵂᴹ or darkᵂᴹ, ψ not.exists.

>> I think that WM thinks that, in our telling,
>> one gets to ω
>> by going out out out to "infinity"
>
> 1, 2, 3, ... ω, ω + 1, ... [Cantor, 1882]

I read your response as a confirmation of
what I think what you think we think.

We (not.WM) do not think that.
Above, you can see reasons we do not think that.

"Infinite" is different.
"Infinite" is not simply "humongous" in dress.up.

Moebius schrieb:

> Am 29.03.2024 um 17:31 schrieb Tom Bola:
>
>> Sets are not ordered, so <bla>
>
> Check: https://en.wikipedia.org/wiki/Partially_ordered_set
> and: https://mathworld.wolfram.com/OrderedSet.html

By default, sets are not ordered.

https://de.wikipedia.org/wiki/Menge_(Mathematik)
Beim Begriff der Menge bleibt außer Betracht, ob es unter den Elementen
zusätzlich irgendeine Ordnung geben könnte, Mengen sind zunächst ungeordnete
Gebilde. Ist eine Reihenfolge der Elemente von Bedeutung, dann spricht man
stattdessen von einer endlichen oder unendlichen Folge, wenn sich die
Folgenglieder mit den natürlichen Zahlen aufzählen lassen (das erste,
das zweite usw.).

https://en.wikipedia.org/wiki/Set_(mathematics)
In a set, all that matters is whether each element is in it or not,
so the ordering of the elements in roster notation is irrelevant (in
contrast, in a sequence, a tuple, or a permutation of a set, the ordering
of the terms matters).

On 3/31/24 8:15 AM, WM wrote:
> Le 30/03/2024 à 19:47, Richard Damon a écrit :
>> On 3/30/24 12:01 PM, WM wrote:
>>> Le 29/03/2024 à 14:29, Richard Damon a écrit :
>>>
>>>> And, it can be shown that ℕ and ℚ are the same size.
>>>
>>> That is believed by some stupids who are too dense to understand
>>> logic. The Os in the matrix
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> ..
>>> cannot disappear by exchange with Xs.
>>
>> So?
>>
>> That isn't the Bijection we are looking for.
>
> It is true for every mapping.

Nope, only if you try to biject WITH a set, and not between two
DIFFERENT sets.

The fact you can't tell the difference, tells us something about you
ability to do logic.

>
> Regards, WM
>
>

On 3/31/2024 1:38 PM, Jim Burns wrote:
> On 3/31/2024 8:20 AM, WM wrote:

>>> [...]
>> [...]
>
> No totally.ordered ⟦0,φ⦆ ⟦0,φ+1⦆ exist such that
>  every non-empty subset of ⟦0,φ⦆ has both
>  a least and a greatest element in the subset
> and
>  NOT every non-empty subset of ⟦0,φ+1⦆ has both
>  a least and a greatest element in the subset.

Still correct, but
I goofed on the definition of ⟦0,φ⦆ etc

⟦0,φ⦆ = {λ ordinal: 0 ≤ λ < φ }

φ ∈ ⟦0,φ⦆ ∌ φ+1

Corrected:
....because,
for each non.empty Sᵩ₊₁ ⊆ ⟦0,φ+1⦆
either
Sᵩ₊₁ = {φ} which has a least and a greatest
or
non.empty Sᵩ₊₁\{φ} ⊆ ⟦0,φ⦆
Sᵩ₊₁\{φ} has a least and a greatest
Sᵩ₊₁ has a least and a greatest

No ordinal ψ exists such that ψ+1 = ω

....because,
if ψ exists, ψ is finite or infinite.
If ψ is finite, then ω = ψ+1 is finite,
but ω is not finite.
If ψ is infinite, then ω > ψ is not first.infinite.
but ω is first.infinite.

ψ is not finite and not infinite.

ψ not.exists such that ψ+1 = ω
Visibleᵂᴹ or darkᵂᴹ, ψ not.exists.

>>> I think that WM thinks that, in our telling,
>>> one gets to ω
>>> by going out out out to "infinity"
>>
>> 1, 2, 3, ... ω, ω + 1, ... [Cantor, 1882]
>
> I read your response as a confirmation of
> what I think what you think we think.
>
> We (not.WM) do not think that.
> Above, you can see reasons we do not think that.
>
> "Infinite" is different.
> "Infinite" is not simply "humongous" in dress.up.

On 3/31/2024 3:37 PM, Jim Burns wrote:
> On 3/31/2024 1:38 PM, Jim Burns wrote:
>> On 3/31/2024 8:20 AM, WM wrote:

>>>> [...]
>>> [...]
>
> ⟦0,φ⦆ = {λ ordinal: 0 ≤ λ < φ }
>
> φ ∈ ⟦0,φ⦆ ∌ φ+1

Dammit.
φ ∈ ⟦0,φ+1⦆ ∌ φ+1

On 3/31/2024 5:15 AM, WM wrote:
> Le 30/03/2024 à 19:47, Richard Damon a écrit :
>> On 3/30/24 12:01 PM, WM wrote:
>>> Le 29/03/2024 à 14:29, Richard Damon a écrit :
>>>
>>>> And, it can be shown that ℕ and ℚ are the same size.
>>>
>>> That is believed by some stupids who are too dense to understand
>>> logic. The Os in the matrix
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> ..
>>> cannot disappear by exchange with Xs.
>>
>> So?
>>
>> That isn't the Bijection we are looking for.
>
> It is true for every mapping.

No true. That would be akin to saying every map is conformal. Totally wrong.

On 3/31/2024 10:14 AM, Moebius wrote:
> Am 30.03.2024 um 17:09 schrieb WM:
>
>> If there are all points on the real line permanently existing, then
>> there is a point next to zero.
>
> No, there is no point "next to zero".
>
> If x is a point "close to zero", then the point x/2 is even closer to
> zero, Du dummer Spinner.

Right. It kind of seems like WM thinks that any number that _he_ cannot
think of must be dark? Humm... ;^o

>
> Geh doch mal zum Psychiater, Mückenheim!

On 03/31/2024 08:31 AM, Tom Bola wrote:
> Ross Finlayson schrieb:
>
>> On 03/31/2024 06:19 AM, Tom Bola wrote:
>>> WM schrieb:
>>>
>>>> Le 30/03/2024 à 20:04, Jim Burns a écrit :
>>>>
>>>>> Hold back just a moment on assigning labels.
>>>>> Is this a fair description of
>>>>> what you think we think?
>>>>>
>>>>> I think that WM thinks that, in our telling,
>>>>> one gets to ω
>>>>> by going out out out to "infinity"
>>>>
>>>> 1, 2, 3, ... ω, ω + 1, ... [Cantor, 1882]
>>>
>>> This is a sequence of all ordinals including limit ordinals, while your
>>> attempt wants to make statements about the natural numbers 1, 2, 3, ...
>>>
>>
>> Just think of it as a model of modularity,
>> that modules are of equal size, that modules
>> have a grain or equi-partitioning, as the
>> number or count of those increases, and
>> then as in the infinite limit and continuum limit,
>> that the boundaries of the modules maintain their
>> proportion, of course though this is still very "natural".
>>
>> The modularity of numbers is a key term with trichotomy.
>>
>> Otherwise this sort of f(n) = n, f(n) = 2n, and so on,
>> simply reflect as a function and functional relation,
>> how it so happens that systems of equations include
>> the algebra of functions, keeping things simple.
>>
>> Whether natural numbers include infinite numbers,
>> the infinitely-grand, has that according to naive
>> expansion of comprehension they do. It's just the
>> regular ordinary opinion of regular ordinary set
>> theory, they don't.
>>
>> There are others, ....
>>
>> Es gibt anderen, ....
>>
>>
>> "Sein ein Hengst, nicht ein Zwerg."
>
> Where do you see trichotomy? The above sequence lists all existent
> limit ordinals (including limit ordinals) winthin a linear order
> up to eps_0, does it.
>

First, it is the number line,
it's numbers, it's a line, it's the number line, trichotomy,
there are the modules apiece, and, their order, makes trichotomy.

Then, an ordering like 1n, 2n makes a trichotomy.

The ordering theory for ordinals and set theory for sets,
and models of counting for accumulation and aggregation,
and various things each primary in their own theories,
like ordering theory, set theory, and number theory.

Ordinals have a trichotomy.

Danke fur schriebern / keine Beleidigung beabstichticht / MfG / ES

Frohe Ostern

Le 30/03/2024 à 23:53, Moebius a écrit :

> Hint: The number of elements in E is aleph_0. And 2 * aleph_0 = aleph_0
> + aleph_0 = aleph_0.

That is nonsense, because one number more than ℕ is one number more,
independent of what alephs can count.
But it is impossible to have one natural number more than all of ℕ.

Regards, WM

Le 31/03/2024 à 17:14, Moebius a écrit :
> Am 30.03.2024 um 17:09 schrieb WM:
>
>> If there are all points on the real line permanently existing, then
>> there is a point next to zero.
>
> No, there is no point "next to zero".

If there are all, then there is one next to zero.
>
> If x is a point "close to zero", then the point x/2 is even closer to
> zero,

Not for all dark points x does x/2 exist.

Regards, WM

Le 31/03/2024 à 17:38, Jim Burns a écrit :
> On 3/31/2024 8:20 AM, WM wrote:

> ψ not.exists such that ψ+1 = ω

If all numbers exist, then also this number exists.
What else should happen by subtracting 1 from ω?
>
>>> I think that WM thinks that, in our telling,
>>> one gets to ω
>>> by going out out out to "infinity"
>>
>> 1, 2, 3, ... ω, ω + 1, ... [Cantor, 1882]
>
> I read your response as a confirmation of
> what I think what you think we think.

Cantor thinks so.

Regards, WM

Le 31/03/2024 à 20:18, Richard Damon a écrit :
> On 3/31/24 8:15 AM, WM wrote:

>>>>> And, it can be shown that ℕ and ℚ are the same size.
>>>>
>>>> That is believed by some stupids who are too dense to understand
>>>> logic. The Os in the matrix
>>>> XOOO...
>>>> XOOO...
>>>> XOOO...
>>>> XOOO...
>>>> ..
>>>> cannot disappear by exchange with Xs.
>>>
>>> So?
>>>
>>> That isn't the Bijection we are looking for.
>>
>> It is true for every mapping.
>
> Nope, only if you try to biject WITH a set, and not between two
> DIFFERENT sets.

The different sets are ℕ and ℚ. The bijection with the first column
does not change that.

Regards, WM

WM wrote on 4/1/2024 :
> Le 31/03/2024 à 20:18, Richard Damon a écrit :
>> On 3/31/24 8:15 AM, WM wrote:
>
>>>>>> And, it can be shown that ℕ and ℚ are the same size.
>>>>>
>>>>> That is believed by some stupids who are too dense to understand logic.
>>>>> The Os in the matrix
>>>>> XOOO...
>>>>> XOOO...
>>>>> XOOO...
>>>>> XOOO...
>>>>> ..
>>>>> cannot disappear by exchange with Xs.
>>>>
>>>> So?
>>>>
>>>> That isn't the Bijection we are looking for.
>>>
>>> It is true for every mapping.
>>
>> Nope, only if you try to biject WITH a set, and not between two DIFFERENT
>> sets.
>
> The different sets are ℕ and ℚ. The bijection with the first column does not
> change that.

So, that's not the way to show it.

Am 01.04.2024 um 17:34 schrieb WM:
> Le 31/03/2024 à 17:38, Jim Burns a écrit :
>> On 3/31/2024 8:20 AM, WM wrote:
>>
>> [No ordinal} ψ exists such that ψ+1 = ω
>>
> If all [ordinal] numbers exist, then also this number exists.

Nope.

Hint: There are finite and infinite ordinals. The finite ordinals are
the elements in IN. Now for NO ψ in IN: ψ + 1 = ω (since for all ψ in
IN: ψ + 1 in IN, but ω is not IN). On the other hand, ω is defined as
the smallest infinite ordinal, hence there is no infnite ordinal ψ such
that ψ + 1 = ω (since ψ would have to be smaller than ψ for ψ + 1 = ω to
hold). Hence there is NO ordinal ψ such that ψ + 1 = ω. qed

> What else should happen by subtracting 1 from ω?

Diese Operation ist nicht definiert, Du Depp!