Re: Contradiction of bijections as a measure for infinite sets

Le 05/04/2024 à 19:03, Jim Burns a écrit :

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

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.

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?

There are no Mückenheim sets. 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, ..
and multgiply 0, 1, 2, 3, ..., ω, by 2. What is the fate of the distance between ℕ and ω? Does it grow to the complete interval ω + 1, ..., ω + k, ..., ω + ω?
Regards, WM