Re: The set of necessary FISONs

On 1/31/2025 6:08 AM, WM wrote:

On 31.01.2025 11:43, FromTheRafters wrote:

WM explained :

On 31.01.2025 11:02, FromTheRafters wrote:

Ordinals aren't guaranteed to increase.

>
What ordinal has not a greater successor?
Ordinals n are well-ordered.
The distance from 0 is n - 0 = n.

>
Starting with omega,
they are all countably infinite
until you reach the uncountably infinite.

>
ω < ω + 1, < ω + 2 < ... < ω + ω = ω2 < ω2 + 1 < ...

[0,0) ⊆ [0,1) ⊆ [0,2) ⊆ ... ⊆ [0,ω)
[0,ω) ⊆ [0,ω+1) ⊆ [0,ω+2) ⊆ ... ⊆ [0,ω+ω) =
[0,ω⋅2) ⊆ [0,ω⋅2+1) ⊆ [0,ω⋅2+2) ⊆ ... ⊆ [0,ω⋅3)
[0,ω⋅3) ⊆ ... ⊆ [0,ω⋅4) ⊆ ... ⊆ [0,ω⋅ω) =
[0,ω^2) ⊆ ... ⊆ [0,ω^3) ⊆ ... ⊆ [0,ω^ω) =
[0,ω^^2) ⊆ ... ⊆ [0,ω^^3) ⊆ ... ⊆ [0,ω^^ω) =
[0,ω^^^2) ⊆ ... ⊆ [0,ω^^^^2) ⊆ ... ⊆ [0,Ω)
[0,Ω) ⊆ [0,Ω+1) ⊆ [0,Ω+2) ⊆ ... ⊆ [0,Ω+ω)
#[0,0) < #[0,1) < #[0,2) < ... < #[0,ω)
#[0,ω) = #[0,ω+1) = #[0,ω+2) = ... = #[0,ω+ω) =
#[0,ω⋅2) = #[0,ω⋅2+1) = #[0,ω⋅2+2) = ... = #[0,ω⋅3)
#[0,ω⋅3) = ... = #[0,ω⋅4) = ... = #[0,ω⋅ω) =
#[0,ω^2) = ... = #[0,ω^3) = ... = #[0,ω^ω) =
#[0,ω^^2) = ... = #[0,ω^^3) = ... = #[0,ω^^ω) =
#[0,ω^^^2) = ... = #[0,ω^^^^2) = ... = ... < #[0,Ω)
#[0,Ω) = #[0,Ω+1) = #[0,Ω+2) = ... = #[0,Ω+ω)
Requiring #[0,k) < #[0,k+1)
restricts k to the first row < ω