On 4/6/2024 9:44 AM, WM wrote:
Le 05/04/2024 à 19:03, Jim Burns a écrit :
On 4/5/2024 5:06 AM, WM wrote:
Dark ordinals reach till ω.
Agreed?
>
⟦0,ξ⟧ which reaches 'til ω both
is a Mückenheim.set and is not a Mückenheim.set.
Agreed?
>
There are no Mückenheim sets.
It looks like you mean to say
there are no not.Mückenheim sets.
Mückenheim.ness is a property which you use
to prove bijections aren't bijections.
Paraphrasing:
| Look!
| This is a not.bijection.
| Therefore,
| that bijection is a not.bijection.
| Because darkᵂᴹ numbers.
You (WM) implicitly use a claim that
all sets are Mückenheim
That implicit claim is what you call logicᵂᴹ.
Mückenheim sets aren't a problem
for you or for us.
We call them finiteⁿᵒᵗᐧᵂᴹ.
You dislike not.Mückenheim sets,
which, de gustibus non disputandum est,
is not a problem, either.
If you don't want to talk about not.Mückenheim sets,
you can not.talk about not.Mückenheim sets.
Your problem is that you think that, by both
disliking and talking about not.Mückenheim sets,
not.Mückenheim sets become Mückenheim sets.
Sadly for you, fortunately for not.you,
arithmetic is not impressed by your résumé.
But we can use the ordinal axis
as Cantor has described it
0, 1, 2, 3, ...,
ω, ω + 1, ..., ω + k, ...,
ω + ω (= ω2), ω2 + 1, ..
The Mückenheim ordinals are confined to
the first row, before ω
We can talk about the others.
We can not.talk about the others.
Either way, the others aren't Mückenheim.
and multiply 0, 1, 2, 3, ..., ω, by 2.
Two times anything on the first row
is something on the first row.
Being on the first row is determined by
the Mückenheim property.
If ⟨1,...,k⟩ has the Mückenheim property
then ⟨1,...,k,k+1,...,k+k⟩ also has it
and is also on the first row, before ω
One way to describe ordinal k as Mückenheim ==
as first.row number k == as k before ω is that
non.0 k and each non.0 before k
has an immediate predecessor.
k+k ̄⟶ k+k-1 ̄⟶ ... ̄⟶ k+1 ̄⟶ k ̄⟶ ... ̄⟶ 0
Each non.0 Mückenheim ordinal (before ω) has
an immediate predecessor.
If ω also had an immediate predecessor,
ω would be on the first row,
instead of beginning the second row.
The difference between ⟦0,m+1⟧ and ⟦0,ω⟧ is
how large?
>
Is it ω for every m?
>
∀k,m ∈ ⟦0,ω⦆: k+m ∈ ⟦0,ω⦆
So, yes.
>
Then it is variable, not a fixed number.
Actually infinite sets are constant.
Potentially infinite collections are variable.
⟦0,ω⦆ doesn't change.
∀k,m ∈ ⟦0,ω⦆: k+m ∈ ⟦0,ω⦆
Perhaps there is a problem with
your assertion that all sets are Mückenheim.
V
Re: V