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

On 4/30/2024 9:12 AM, WM wrote:

Le 28/04/2024 à 19:29, Jim Burns a écrit :

1.
We describe some things.
For our convenience,
we refer to what we describe by some label.
Whatever label we use, "natural number" or
"flying rainbow sparkle pony",
using that label uses _that description_

If n can be counted to from 0
then n⋅2 can be counted to _from n_
then n⋅2 can be counted to from 0 _through n_

>
That is true.
>

If n is before ω
then n⋅2 before ω

>
That is not true.

If any number below n canNOT be counted.to from 0
then n itself canNOT be counted.to from 0
For our convenience,
I use 澒 to label that thing which is
the first.upper.bound of
numbers which can be counted.to from 0
Consider a 澒.crossing pair ⟨j,k⟩ of numbers.
j < 澒 < k
k canNOT be counted.to from 0
j CAN be counted.to from 0
k canNOT be counted.to from 0  because,
if k CAN be counted.to from 0
then
澒 is NOT an upper.bound of CAN.numbers
j CAN be counted.to from 0  because,
if j canNOT be counted.to from 0
then
no number after j CAN be counted.to from 0
j is an upper.bound of CAN.numbers
j < 澒 is an upper.bound before 澒
澒 is NOT the first.upper.bound of CAN.numbers.
However,
澒 IS the first.upper.bound of CAN.numbers.
If n is before 澒
then n⋅2 is before 澒
...because
⟨n,n⋅2⟩ is NEVER a 澒.crossing pair
Either n AND n⋅2 CAN be counted.to from 0
or n AND n⋅2 canNOT be counted.to from 0
But, in a a 澒.crossing pair
the first CAN and the second canNOT.