Re: because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

On 4/22/2024 2:48 PM, Moebius wrote:

Am 22.04.2024 um 19:32 schrieb Jim Burns:

⦅0,ω+ω⦆ fits ⦅0,ω⦆

>
Def:
(u, v) := {x e ORD : u < x < v}
(with x,y e ORD)
>
Then ⦅0,ω⦆ c ⦅0,ω+ω⦆, but not ⦅0,ω+ω⦆ c ⦅0,ω⦆.
>
After all, ω < ω+1 < ω+2 < ... < ω+ω.
>
So with "I fits J",
you don't mean "I fits in J",
but rather "J fits in I", I guess.
Right?
>
A commen term in this context is "contains".
Hence ⦅0,ω+ω⦆ contains ⦅0,ω⦆.

I am using "fit" differently.
X fits Y  ⟺  X [≤] Y  ⟺
exists 1.to.1 f: X ⇉ Y
Yes
ω < ω+1 < ω+2 < ... < ω+ω
However,
ω [≤≥] ω+1 [≤≥] ω+2 [≤≥] ... [≤≥] ω+ω
ω [=] ω+1 [=] ω+2 [=] ... [=] ω+ω
The distinction I make for WM is
κ [≤≱] κ+1 [≤≱] ... ω [≤≥] ω+1 [≤≥] ...
κ [<] κ+1 [<] ... ω [=] ω+1 [=] ...
I'm sorry for the confusion.
I am trying to shorten my posts enough that
WM will read them.
I am an eternal optimist.
The stylistic choice I make is a two.edged sword.
I feel it is a dynamic, very.readable style,
leaning toward the Ernest.Hemingway.ish.
However, deleting context courts confusion,
as you have noticed.