Re: Contradiction of bijections as a measure for infinite sets

On 4/5/2024 5:06 AM, WM wrote:

Le 04/04/2024 à 17:03, Jim Burns a écrit :

always ⟦0,m+1⟧ ≠ ⟦0,ω⟧

I define a Mückenheim set such that,
for any set,  either
all injections are onto,  or
all injections are not.onto.
Mückenheim.set M  ⟺
∀S: ∀f:M⇉S=f(M) ∨ ∀g:M⇉S≠g(M)
not.Mückenheim.set M  ⟺
∃S: ∃f:M⇉S≠f(M) ∧ ∃g:M⇉S=g(M)
Define ⟦0,ω⦆ = lub{Mückenheim ⟦0,m⟧}

The difference between ⟦0,m+1⟧ and ⟦0,ω⟧ is
how large?

Larger than any Mückenheim ⟦0,m⟧
  thus
not a Mückenheim.set
  thus
the same as
between ⟦0,m⟧ and ⟦0,ω⟧  and
between ⟦0,m+2⟧ and ⟦0,ω⟧.

Is it ω for every m?

∀k,m ∈ ⟦0,ω⦆: k+m ∈ ⟦0,ω⦆
So, yes.

Then what are the ordinals between m and ω?

They are the elements of ⦅m,ω⦆

They are dark.

Visibleᵂᴹ or darkᵂᴹ, always ⟦0,m+1⟧ ≠ ⟦0,ω⟧

On the other hand
no ordinal fits between ℕ and ω.

ℕ = ⟦0,ω⦆
So, we agree on something.

Dark ordinals reach till ω.
Agreed?

⟦0,ξ⟧ which reaches 'til ω  both
is a Mückenheim.set and is not a Mückenheim.set.
Agreed?