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

Le 20/04/2024 à 00:24, Jim Burns a écrit :

On 4/19/2024 11:15 AM, WM wrote:

Le 19/04/2024 à 00:09, Jim Burns a écrit :

On 4/18/2024 10:54 AM, WM wrote:

 

It is enough if you explain
where ω is in the lower line:

>
0, 1, 2, 3, ...,   ω
|  |  |  |  |||    |
0, 2, 4, 6, ..., ω*2
>

∀κ < ω: k⋅2 < ω

>
That means
the space between ω and ω*2 remains empty of
poducts 2k, and
not all natural numbers have been doubled
because new products have been inserted below ω.

 Between ω and ω⋅2 is empty of
products k⋅2 from k < ω

>
How do the ordinal numbers ω+1, ω+2, ... come into being?

Nothing is inserted anywhere.
 Each finite even is finite and below ω
and is double an ordinal finite and below ω

>
Then that one directly before ω is not multiplied. Or it is not existing. But what exists directly before ω?