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

On 4/25/2024 4:03 PM, WM wrote:

Le 24/04/2024 à 21:42, Jim Burns a écrit :

On 4/24/2024 12:57 PM, WM wrote:

Le 23/04/2024 à 21:55, Jim Burns a écrit :

Arithmetic of the familiar.

>
Nevertheless it is wrong because
for every set {1, 2, 3, ..., n}
doubling extends the set.

>
Doubling doesn't extend it to or beyond ω

>
If all smaller numbers are doubled,
then there is no place for
the doubled numbers below ω.

"Infinite" does not mean "humongous".
finite ⟺ can be counted to from 0
finite ⟺ below ω
If n is below ω
then 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
then n⋅2 is below ω
If n is below w
then n⋅2 is below w

If all smaller numbers are doubled,
then there is no place for
the doubled numbers below ω.
>

The arithmetic of ω is not
the familiar arithmetic.

>
The arithmetic of doubling produces
other numbers than the doubled ones.

The arithmetic of doubling
numbers which can be counted to
produces only
doubled numbers which can be counted to.
"Infinite" does not mean "humongous".

ω+i ⟼ 2⋅i

>
ω+i is not mapped.

>
It is mapped in front of your eyes, sic: '⟼'
not your doubling.map, but
the map which show that ⦅0,ω⋅2⦆ fits ⦅0,ω⦆

>
You claim that ω+3 = 2*3?

 I defined a map for which ω+3 ⟼ 2⋅3
 It would explain a lot of your (WM's) posts,
if you don't know what a function/map is,
what familiar addition is,
what familiar multiplication is.
>
It would also raise the question of
who is responsible for putting you (WM)
in front of a classroom of students.

>
Those are experts which
a disappearing Bob cannot estimate.

Experts who think, as you (WM) do,
that ω+3 ⟼ 2⋅3 means ω+3 = 2⋅3
That is a radical new use of the word "expert".