On 3/28/2024 5:40 PM, WM wrote:
> Le 28/03/2024 à 21:25, Jim Burns a écrit :
>> On 3/27/2024 5:50 PM, WM wrote:
>>> Le 27/03/2024 à 15:23, FromTheRafters a écrit :

>>>> Not showing a bijection is not the same as
>>>> not having a bijection.
>>>
>>> Showing a not bijection proves
>>> different sizes of sets.
>>
>> I'll call that a Mückenheim.set.
>
>> Mückenheim.sets are finiteⁿᵒᵗᐧᵂᴹ sets.
>
> No.
> The set {1, 2, 3, 4, 5, ..., ω} is infinite,
> although there is a last element.

The set {1, 2, 3, 4, 5, ..., ω} is not.Mückenheim.

Consider not.bijection
g: {1,2,3,4,5,...,ω} ⇉ {1,2,3,4,5,...,ω}
g(ω) = 1
g(n) = n+1 otherwise

{1,2,3,4,5,...,ω} is not a different size than
{1,2,3,4,5,...,ω}

finiteⁿᵒᵗᐧᵂᴹ does not mean what you think it means.

> Under f(x) = 2x
> we get the image {2, 4, 6, 8, 10, ..., 2ω].
> The mapping restricted to the natural numbers shows
> less evens than naturals.

s/less/fewer

No.
f(x) = 2x
not.bijection f: ℕ ⇉ ℕ
f(ℕ) = 𝔼 ≠ ℕ

If Mückenheim ℕ then |ℕ| ≠ |ℕ|

not.Mückenheim ℕ

One can't draw valid conclusions from
the assertion that ℕ is Mückenheim.

>>> But we can assume that
>>> very simple mappings like f(x) = x are true
>>> even for dark elements.
>>> Therefore
>>> between the rational numbers and the natural numbers
>>> f(n) = n/1 can be accepted,
>>> also f(n) = 1/n,
>>> but not f(n) = 2n.
>>
>> Where are darkᵂᴹ numbers introduced?
>
> They are the places where
> Bob rests when the mapping is finished.

Where in these definitions
did I introduce darkᵂᴹ numbers?

For k,m ∈ ℕᴶᵛᴺ define addition such that
k+m = nₖₘ ⟺
exists Mückenheim.3.tuple.sequence ⟨⟨k,0,k⟩,…,⟨k,m,nₖₘ⟩⟩
such that
⟨k,0,k⟩ ∈ ⟨⟨k,0,k⟩,…,⟨k,m,nₖₘ⟩⟩ ∧
⟨k,m,nₖₘ⟩ ∈ ⟨⟨k,0,k⟩,…,⟨k,m,nₖₘ⟩⟩ ∧
∀i ∈ m: ∃!⟨k,i,j⟩ ∈ ⟨⟨k,0,k⟩,…,⟨k,m,nₖₘ⟩⟩ ∧
∀⟨k,i,j⟩ ∈ ⟨⟨k,0,k⟩,…,⟨k,m,nₖₘ⟩⟩ ∋ ⟨k,i⁺¹,j⁺¹⟩

lemma. nₖₘ ∈ ℕᴶᵛᴺ

For k,m ∈ ℕᴶᵛᴺ define multiplication such that
k⋅m = n′ₖₘ ⟺
exists Mückenheim.3.tuple.sequence ⟨⟨k,0,0⟩,…,⟨k,m,n′ₖₘ⟩⟩
such that
⟨k,0,0⟩ ∈ ⟨⟨k,0,0⟩,…,⟨k,m,n′ₖₘ⟩⟩ ∧
⟨k,m,n′ₖₘ⟩ ∈ ⟨⟨k,0,0⟩,…,⟨k,m,n′ₖₘ⟩⟩ ∧
∀i ∈ m: ∃!⟨k,i,j⟩ ∈ ⟨⟨k,0,0⟩,…,⟨k,m,n′ₖₘ⟩⟩ ∧
∀⟨k,i,j⟩ ∈ ⟨⟨k,0,0⟩,…,⟨k,m,n′ₖₘ⟩⟩ ∋ ⟨k,i⁺¹,j+k⟩

lemma. n′ₖₘ ∈ ℕᴶᵛᴺ

For k ∈ ℕᴶᵛᴺ define fraction iₖ/jₖ such that
sₖ = max{h: (h-1)(h-2)/2 < k }
iₖ = k-(sₖ-1)(sₖ-2)/2
jₖ = sₖ-iₖ

lemma. iₖ,jₖ ∈ ℕᴶᵛᴺ

For i,j ∈ ℕᴶᵛᴺ define index kᵢⱼ such that
sᵢⱼ = i+j
kᵢⱼ = (sᵢⱼ-1)(sᵢⱼ)-2)/2+i

lemma. kᵢⱼ ∈ ℕᴶᵛᴺ

lemma. kᵢⱼ = k ⟺ ⟨i,j⟩ = ⟨iₖ,jₖ⟩

On 3/28/2024 2:40 PM, WM wrote:
> Le 28/03/2024 à 21:25, Jim Burns a écrit :
>> On 3/27/2024 5:50 PM, WM wrote:
>>> Le 27/03/2024 à 15:23, FromTheRafters a écrit :
>>
>>>> Not showing a bijection is not the same as
>>>> not having a bijection.
>>>
>>> Showing a not bijection proves
>>> different sizes of sets.
>>
>> I'll call that a Mückenheim.set.
>
>> Mückenheim.sets are finiteⁿᵒᵗᐧᵂᴹ sets.
>
> No. The set  {1, 2, 3, 4, 5, ..., ω} is infinite, although there is a
> last element.

Huh? If it has a last element its NOT infinite...

> Under f(x) = 2x we get the image {2, 4, 6, 8, 10, ..., 2ω]. The mapping
> restricted to the natural numbers shows less evens than naturals.
>
>>> But we can assume that
>>> very simple mappings like f(x) = x are true
>>> even for dark elements.
>>> Therefore
>>> between the rational numbers and the natural numbers
>>> f(n) = n/1 can be accepted,
>>> also f(n) = 1/n,
>>> but not f(n) = 2n.
>>
>> Where are darkᵂᴹ numbers introduced?
>
> They are the places where Bob rests when the mapping is finished.
>
> Regards, WM
>
>

On 3/28/2024 5:54 PM, Chris M. Thomasson wrote:
> On 3/28/2024 2:40 PM, WM wrote:
>> Le 28/03/2024 à 21:25, Jim Burns a écrit :
>>> On 3/27/2024 5:50 PM, WM wrote:
>>>> Le 27/03/2024 à 15:23, FromTheRafters a écrit :
>>>
>>>>> Not showing a bijection is not the same as
>>>>> not having a bijection.
>>>>
>>>> Showing a not bijection proves
>>>> different sizes of sets.
>>>
>>> I'll call that a Mückenheim.set.
>>
>>> Mückenheim.sets are finiteⁿᵒᵗᐧᵂᴹ sets.
>>
>> No. The set  {1, 2, 3, 4, 5, ..., ω} is infinite, although there is a
>> last element.
>
> Huh? If it has a last element its NOT infinite...

Oh I missed something, sorry. Don't try to tell me that ω is your
infamous largest natural number... ?

Think of a line with points p0 and p1. Draw a line from p0 to p1. We
just covered an infinite number of points in between...

>> Under f(x) = 2x we get the image {2, 4, 6, 8, 10, ..., 2ω]. The
>> mapping restricted to the natural numbers shows less evens than naturals.

There are an infinite number of even naturals. There are an infinite
number of naturals. They both have an infinite number of items, so to speak.

>>
>>>> But we can assume that
>>>> very simple mappings like f(x) = x are true
>>>> even for dark elements.
>>>> Therefore
>>>> between the rational numbers and the natural numbers
>>>> f(n) = n/1 can be accepted,
>>>> also f(n) = 1/n,
>>>> but not f(n) = 2n.
>>>
>>> Where are darkᵂᴹ numbers introduced?
>>
>> They are the places where Bob rests when the mapping is finished.
>>
>> Regards, WM
>>
>>
>

On 3/28/2024 8:54 PM, Chris M. Thomasson wrote:
> On 3/28/2024 2:40 PM, WM wrote:
>> Le 28/03/2024 à 21:25, Jim Burns a écrit :
>>> On 3/27/2024 5:50 PM, WM wrote:
>>>> Le 27/03/2024 à 15:23, FromTheRafters a écrit :

>>>>> Not showing a bijection is not the same as
>>>>> not having a bijection.
>>>>
>>>> Showing a not bijection proves
>>>> different sizes of sets.
>>>
>>> I'll call that a Mückenheim.set.
>>
>>> Mückenheim.sets are finiteⁿᵒᵗᐧᵂᴹ sets.
>>
>> No. The set {1, 2, 3, 4, 5, ..., ω} is infinite,
>> although there is a last element.
>
> Huh?
> If it has a last element its NOT infinite...

No,
the same set has transitive.trichotomous.orders
1 < 2 < 3 < 4 < ... < ω
and
ω <′ 1 <′ 2 <′ 3 <′ 4 <′ ...
and
.... <″ 4 <″ 2 <″ ω <″ 1 <″ 3 <″ ...

The same set is infiniteⁿᵒᵗᐧᵂᴹ in each order.

What < <′ <″ have in common is
nonempty subsets without two ends.
1 < 2 < 3 < 4 < ...
2 < 3 < 4 < 5 < ...
3 < 4 < 5 < 6 < ...

ω <′ 1 <′ 2 <′ 3 <′ 4 <′ ...
1 <′ 2 <′ 3 <′ 4 <′ 5 <′ ...
2 <′ 3 <′ 4 <′ 5 <′ 6 <′ ...

.... <″ 4 <″ 2 <″ ω <″ 1 <″ 3 <″ ...
ω <″ 1 <″ 3 <″ 5 <″ 7 <″ ...
.... <″ 8 <″ 6 <″ 4 <″ 2 <″ ω

For an infiniteⁿᵒᵗᐧᵂᴹ set
_each transitive.trichotomous.order_
has nonempty non.two.ended subsets.

For a finiteⁿᵒᵗᐧᵂᴹ set
_each transitive.trichotomous.order_
does not have nonempty non.two.ended subsets.

No set exists
between finiteⁿᵒᵗᐧᵂᴹ and infiniteⁿᵒᵗᐧᵂᴹ with
some orders with non.two.ended subsets and
some orders without non.two.ended subsets.

For each set,
all orders are all.sub.two.ended or
no orders are all.sub.two.ended.

On 3/28/24 4:07 PM, WM wrote:
> Le 27/03/2024 à 18:54, Jim Burns a écrit :
>> On 3/27/2024 9:38 AM, WM wrote:
>>> Le 26/03/2024 à 16:40, Jim Burns a écrit :
>>
>>>> ℕ and ℚ have the same infinity.
>>>
>>> Only if
>>> logic (every lossless exchange is lossless)
>>> is violated and damaged, i.e.,
>>> only in matheology.
>>
>> ℕ and ℚᶠʳᵃᶜ are the same size.
>
> ℕ and ℕ are the same size.
> ℚ and ℚ are the same size.

And, it can be shown that ℕ and ℚ are the same size.

>
> Removing a proper fraction decreases ℚ but leaves it larger than ℕ. When
> the size changes it cannot remain the same.

Nope, Removing a single element from an infinite set doesn't change its
size.

>
> Showing a not bijection proves different sizes of sets.
> Why is that more meaningful than Cantor's bijections?

Nope, showing NO BIJECTION CAN EXIST proves different sizes.

Having just a failed bijection shows nothing for infinite sets.

>
> Between infinite sets there cannot exist any mapping because most
> elements are dark. But we can assume that very simple mappings like f(x)
> = x are true even for dark elements.

Only because you logic doesn't handle the infinite sets.

Your "Darkness" is just your logic system saying "No" to its misuse.

>
> Therefore between the rational numbers and the natural numbers f(n) =
> n/1 can be accepted, also f(n) = 1/n, but not f(n) = 2n.

Sure it can, you just don't understand it because you are using
imporoper logic for such a system.

Like trying to argue about colors in a black and white photo.

>
> Regards, WM
>
>
> Regards, WM
>
>

On 3/28/24 4:00 PM, WM wrote:
> Le 28/03/2024 à 01:57, Richard Damon a écrit :
>> On 3/27/24 9:57 AM, WM wrote:
>
>>> Only if the bijection was between potentially infinite sets.
>>
> *********************
>> Which N and E are/
> *********************

No, N and E are ACTUALLY infinite sets of potentially infinite numbers
(if I am understanding the meaning of your not well defined terms)

>>
>>> I understand that your claims can only be satisfied by potential
>>> infinity.
>
> *******************************************************************
>> We are talking about ACTUAL infinite sets, not just "potential",
> *******************************************************************

Right, so FINITE logic doesn't apply to them.

>
> Regards, WM

The clown WM drivels:

> The set {1, 2, 3, 4, 5, ..., ω} is infinite, although there is a
> last element.

One may construct such a set "N-with-omega".

> Under f(x) = 2x we get the image {2, 4, 6, 8, 10, ..., 2ω}.

One may also define such a function, which domain contains both sort of ordinals,
i.e. natural numbers and also a limit ordinal like ω.

> The mapping restricted to the natural numbers shows less evens than naturals.

The above sentence is very idiotic nonsense, because this set would contain
the union of W1 = {2, 4, 6, 8, 10, ...} of card ω and W2 = {2ω} of card 1.

You are way too dense for even simplest thinking and math...

Chris M. Thomasson schrieb:
> On 3/28/2024 2:40 PM, WM wrote:
>> Le 28/03/2024 à 21:25, Jim Burns a écrit :
>>> On 3/27/2024 5:50 PM, WM wrote:
>>>> Le 27/03/2024 à 15:23, FromTheRafters a écrit :
>>>
>>>>> Not showing a bijection is not the same as
>>>>> not having a bijection.
>>>>
>>>> Showing a not bijection proves
>>>> different sizes of sets.
>>>
>>> I'll call that a Mückenheim.set.
>>
>>> Mückenheim.sets are finiteⁿᵒᵗᐧᵂᴹ sets.
>>
>> No. The set  {1, 2, 3, 4, 5, ..., ω} is infinite, although there is a
>> last element.
>
> Huh? If it has a last element its NOT infinite...

Oops, you might also write that this way: { ω, 1, 2, 3, 4, 5, ... }.

Addendum:

> The clown WM drivels:
>
>> The set {1, 2, 3, 4, 5, ..., ω} is infinite, although there is a
>> last element.

Tom Bola wrote:
> One may construct such a set "N-with-omega".

Sets are not ordered, so there cannot be a "last element" in any set, of course.

On 3/28/24 3:55 PM, WM wrote:
> Le 27/03/2024 à 23:37, FromTheRafters a écrit :
>> WM expressed precisely :
>
>>> Showing a not bijection proves different sizes of sets.
>>
>> No it doesn't.
>
> Why not?
>
> Regards, WM

Because it is showable that you CAN make non-working attempts at
bijections on infinite sets that you can also show working bijections,
so clearly, failed bijections do not show that infinite sets are of a
different size.

That is just one of the interesting properties of infinity and
transfinite math.

Some things that might seem "obvious" in the finite world, just don't work.

On 3/28/24 3:57 PM, WM wrote:
> Le 28/03/2024 à 01:59, Richard Damon a écrit :
>> On 3/27/24 5:50 PM, WM wrote:
>
>>> Showing a not bijection proves different sizes of sets.
>>> Why is that more meaningful than Cantor's bijections?
>>
>> Nope. Cantor shows that there may well be "failed" attempts at a
>> bijection that actually don't tell us anything about the sizes of two
>> infinite sets.
>
> Why do you think so?
>
> Regards, WM
>
>

Because it is easy to come up with failed bijections for sets that a
succefful bijection exists.

Thus failed bijections for infinite sets do not prove different sizes.

This is just one of the interesting properties of infinite/transfinite
logic and math. Something you likely can't understand since you are
stuck in finite logic.

On 3/29/2024 7:13 AM, Tom Bola wrote:
> Chris M. Thomasson schrieb:
>> On 3/28/2024 2:40 PM, WM wrote:
>>> Le 28/03/2024 à 21:25, Jim Burns a écrit :
>>>> On 3/27/2024 5:50 PM, WM wrote:
>>>>> Le 27/03/2024 à 15:23, FromTheRafters a écrit :
>>>>
>>>>>> Not showing a bijection is not the same as
>>>>>> not having a bijection.
>>>>>
>>>>> Showing a not bijection proves
>>>>> different sizes of sets.
>>>>
>>>> I'll call that a Mückenheim.set.
>>>
>>>> Mückenheim.sets are finiteⁿᵒᵗᐧᵂᴹ sets.
>>>
>>> No. The set  {1, 2, 3, 4, 5, ..., ω} is infinite, although there is a
>>> last element.
>>
>> Huh? If it has a last element its NOT infinite...
>
> Oops, you might also write that this way: { ω, 1, 2, 3, 4, 5, ... }.

I got confused. I thought that ω was WM's largest natural number, its
last element, so to speak.

On 3/29/2024 2:46 AM, Jim Burns wrote:
> On 3/28/2024 8:54 PM, Chris M. Thomasson wrote:
>> On 3/28/2024 2:40 PM, WM wrote:
>>> Le 28/03/2024 à 21:25, Jim Burns a écrit :
>>>> On 3/27/2024 5:50 PM, WM wrote:
>>>>> Le 27/03/2024 à 15:23, FromTheRafters a écrit :
>
>>>>>> Not showing a bijection is not the same as
>>>>>> not having a bijection.
>>>>>
>>>>> Showing a not bijection proves
>>>>> different sizes of sets.
>>>>
>>>> I'll call that a Mückenheim.set.
>>>
>>>> Mückenheim.sets are finiteⁿᵒᵗᐧᵂᴹ sets.
>>>
>>> No. The set {1, 2, 3, 4, 5, ..., ω} is infinite,
>>> although there is a last element.
>>
>> Huh?
>> If it has a last element its NOT infinite...
>
> No,
> the same set has transitive.trichotomous.orders
[...]

Yeah. Think of two points p0, p1, draw a line, we just covered infinite
points even though it has a defined start and end.

Chris M. Thomasson schrieb:
> On 3/29/2024 7:13 AM, Tom Bola wrote:
>> Chris M. Thomasson schrieb:
>>> On 3/28/2024 2:40 PM, WM wrote:
>>>> Le 28/03/2024 à 21:25, Jim Burns a écrit :
>>>>> On 3/27/2024 5:50 PM, WM wrote:
>>>>>> Le 27/03/2024 à 15:23, FromTheRafters a écrit :
>>>>>
>>>>>>> Not showing a bijection is not the same as
>>>>>>> not having a bijection.
>>>>>>
>>>>>> Showing a not bijection proves
>>>>>> different sizes of sets.
>>>>>
>>>>> I'll call that a Mückenheim.set.
>>>>
>>>>> Mückenheim.sets are finiteⁿᵒᵗᐧᵂᴹ sets.
>>>>
>>>> No. The set  {1, 2, 3, 4, 5, ..., ω} is infinite, although there is a
>>>> last element.
>>>
>>> Huh? If it has a last element its NOT infinite...
>>
>> Oops, you might also write that this way: { ω, 1, 2, 3, 4, 5, ... }.
>
> I got confused. I thought that ω was WM's largest natural number, its
> last element, so to speak.

Sure...

On 3/29/2024 4:29 PM, Tom Bola wrote:
> Chris M. Thomasson schrieb:
>> On 3/29/2024 7:13 AM, Tom Bola wrote:
>>> Chris M. Thomasson schrieb:
>>>> On 3/28/2024 2:40 PM, WM wrote:
>>>>> Le 28/03/2024 à 21:25, Jim Burns a écrit :
>>>>>> On 3/27/2024 5:50 PM, WM wrote:
>>>>>>> Le 27/03/2024 à 15:23, FromTheRafters a écrit :
>>>>>>
>>>>>>>> Not showing a bijection is not the same as
>>>>>>>> not having a bijection.
>>>>>>>
>>>>>>> Showing a not bijection proves
>>>>>>> different sizes of sets.
>>>>>>
>>>>>> I'll call that a Mückenheim.set.
>>>>>
>>>>>> Mückenheim.sets are finiteⁿᵒᵗᐧᵂᴹ sets.
>>>>>
>>>>> No. The set  {1, 2, 3, 4, 5, ..., ω} is infinite, although there is a
>>>>> last element.
>>>>
>>>> Huh? If it has a last element its NOT infinite...
>>>
>>> Oops, you might also write that this way: { ω, 1, 2, 3, 4, 5, ... }.
>>
>> I got confused. I thought that ω was WM's largest natural number, its
>> last element, so to speak.
>
> Sure...

I know for sure that WM thinks there is a largest natural number. That
must be his last element. Still, I do not know why he thinks that way.
Oh well. Shit happens.

On 3/29/2024 7:31 PM, Chris M. Thomasson wrote:
> On 3/29/2024 4:29 PM, Tom Bola wrote:
>> Chris M. Thomasson schrieb:
>>> On 3/29/2024 7:13 AM, Tom Bola wrote:
>>>> Chris M. Thomasson schrieb:
>>>>> On 3/28/2024 2:40 PM, WM wrote:
>>>>>> Le 28/03/2024 à 21:25, Jim Burns a écrit :
>>>>>>> On 3/27/2024 5:50 PM, WM wrote:
>>>>>>>> Le 27/03/2024 à 15:23, FromTheRafters a écrit :

>>>>>>>>> Not showing a bijection is not the same as
>>>>>>>>> not having a bijection.
>>>>>>>>
>>>>>>>> Showing a not bijection proves
>>>>>>>> different sizes of sets.
>>>>>>>
>>>>>>> I'll call that a Mückenheim.set.
>>>>>>
>>>>>>> Mückenheim.sets are finiteⁿᵒᵗᐧᵂᴹ sets.
>>>>>>
>>>>>> No.
>>>>>> The set {1, 2, 3, 4, 5, ..., ω} is infinite,
>>>>>> although there is a last element.
>>>>>
>>>>> Huh? If it has a last element its NOT infinite...

> I know for sure that WM thinks
> there is a largest natural number.
> That must be his last element.

I think that, in WM's telling,
ω is _immediately after_ the largest natural number,
and
proofs that
"immediately before ω" does not describe
anything non.self.contradictory
are proofs that
darkᵂᴹ numbers exist.
In WM's telling.

I think that WM thinks that, in our telling,
ω is a natural number one gets to
by going out out out to "infinity"
and the word "infinite" is in the same family
as "humongous" and "ginormous" except it uses
special characters WM won't or can't read,
which which are what makes "infinite" mathematics.

For the record,
matheologians (AKA not.WM) do not say that.
That which we call ω
by any other word would smell as sweet.

Finite and infinite are as different as
rational and irrational are.

> Still, I do not know why he thinks that way.
> Oh well. Shit happens.

Why he thinks that
is a question for psychologists.

I speculate that
Wolfgang Mückenheim has been seduced by
decades of students at Hochschule Augsburg
memorizing whatever fool thing pops into his head
into thinking that
it is impossible for him to be wrong.
Saying otherwise, or worse, showing otherwise
is just _rude_
like a student interrupting a lecture.
Possibly.

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.
>
> Nope, Removing a single element from an infinite set doesn't change its
> size.

That is believed by some stupids who are too dense to understand logic.
Removing one element makes the set having one element less than before.

Regrads, WM

Le 29/03/2024 à 18:05, Richard Damon a écrit :
> On 3/28/24 3:55 PM, WM wrote:
>> Le 27/03/2024 à 23:37, FromTheRafters a écrit :
>>> WM expressed precisely :
>>
>>>> Showing a not bijection proves different sizes of sets.
>>>
>>> No it doesn't.
>>
>> Why not?
>
> Because it is showable that you CAN make non-working attempts at
> bijections on infinite sets that you can also show working bijections,

Only if the Os in the matrix
XOOO...
XOOO...
XOOO...
XOOO...
...
could disappear by exchange with Xs which is ipossible.

Regards, WM

Le 29/03/2024 à 23:31, "Chris M. Thomasson" a écrit :

> I know for sure that WM thinks there is a largest natural number. That
> must be his last element. Still, I do not know why he thinks that way.

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

Regards, WM

Le 30/03/2024 à 13:56, Jim Burns a écrit :

> I think that, in WM's telling,
> ω is _immediately after_ the largest natural number,

like the smallest unit fraction is existing.

> I think that WM thinks that, in our telling,
> ω is a natural number

No.

> Why he thinks that
> is a question for psychologists.

It is a simple result of the permanent existence of all numbers and
point.For ever n, there is 2n, is potential infinity. You first must give
an n. If however all 2n are there, then none can be created.

Regards, WM

Le 29/03/2024 à 14:09, Tom Bola a écrit :
> Prof. Dr. Mückenheim taught:
>
>> The set {1, 2, 3, 4, 5, ..., ω} is infinite, although there is a
>> last element.
>
> One may construct such a set "N-with-omega".
>
>> Under f(x) = 2x we get the image {2, 4, 6, 8, 10, ..., 2ω}.
>
> One may also define such a function, which domain contains both sort of
> ordinals,
> i.e. natural numbers and also a limit ordinal like ω.
>
>> The mapping restricted to the natural numbers shows less evens than naturals.
>
> this set would contain
> the union of W1 = {2, 4, 6, 8, 10, ...} of card ω and W2 = {2ω} of card 1.
>
No. Cantor accepts ω, ω + 2, ω + 4, ... These numbers can be realized
by repeated addition. Mutiplication is a shorthand of addition.

Regards, WM

The clown WM drivels:

> Tom Bola a écrit :
>> The clown WM drivels:
>>
>>> The set {1, 2, 3, 4, 5, ..., ω} is infinite, although there is a
>>> last element.
>>
>> One may construct such a set "N-with-omega".
>>
>>> Under f(x) = 2x we get the image {2, 4, 6, 8, 10, ..., 2ω}.
>>
>> One may also define such a function, which domain contains both sort of
>> ordinals,
>> i.e. natural numbers and also a limit ordinal like ω.
>>
>>> The mapping restricted to the natural numbers shows less evens than naturals.
>>
>> this set would contain
>> the union of W1 = {2, 4, 6, 8, 10, ...} of card ω and W2 = {2ω} of card 1.
>>
> No. Cantor accepts ω, ω + 2, ω + 4, ... These numbers can be realized
> by repeated addition. Mutiplication is a shorthand of addition.

You complete moron are unable to get that your set {1, 2, 3, 4, 5, ..., ω}
does not belong to the math of natural numbers because ω is the limiting
ordinal of IN = {1, 2, 3, 4, 5, ...}.

Piss off already, blithering retard...

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.

>>
>> Nope, Removing a single element from an infinite set doesn't change
>> its size.
>
> That is believed by some stupids who are too dense to understand logic.
> Removing one element makes the set having one element less than before.
>
> Regrads, WM
>

Which doesn't affect the "size" of infinity.

DEFINITIONS you know.

On 3/30/24 12:05 PM, WM wrote:
> Le 29/03/2024 à 18:05, Richard Damon a écrit :
>> On 3/28/24 3:55 PM, WM wrote:
>>> Le 27/03/2024 à 23:37, FromTheRafters a écrit :
>>>> WM expressed precisely :
>>>
>>>>> Showing a not bijection proves different sizes of sets.
>>>>
>>>> No it doesn't.
>>>
>>> Why not?
>>
>> Because it is showable that you CAN make non-working attempts at
>> bijections on infinite sets that you can also show working bijections,
>
> Only if the Os in the matrix
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ..
> could disappear by exchange with Xs which is ipossible.
>
> Regards, WM
>

Failed bijections do not prove anything for infinite sets.

Since we CAN create a bijection between the sets:

XXXXXXXX....

and

OOO...
OOO...
OOO...
....

We can show they they are the same size, and that you are just ignorant.

On 3/30/2024 12:17 PM, WM wrote:
> Le 30/03/2024 à 13:56, Jim Burns a écrit :

>> I think that, in WM's telling,
>> ω is _immediately after_ the largest natural number,
>
> like the smallest unit fraction is existing.
>
>> I think that WM thinks that, in our telling,
>> ω is a natural number
>
> No.

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"
and the word "infinite" is in the same family
as "humongous" and "ginormous" except it uses
special characters, which which are
what makes "infinite" mathematics.