Re: how

Richard Damon schrieb:

On 4/17/24 2:52 PM, WM wrote:

Le 17/04/2024 à 01:29, Richard Damon a écrit :

On 4/16/24 2:36 PM, WM wrote:

Le 16/04/2024 à 18:38, Mike Terry a écrit :

On 16/04/2024 16:05, WM wrote:

Le 16/04/2024 à 02:35, Mike Terry a écrit :

On 15/04/2024 14:00, Tom Bola wrote:

>

0, 1, 2, 3, ...,   w,   w+1,   w+2,   w+3, ...     w+w
|  |  |  |  |||    |     |      | |            |
0, 2, 4, 6, ..., w*2, w*2+2, w*2+4, w*2+6, ... w*2+w*2

>
Should be:
>
0, 1, 2, 3, ...,   w,   w+1,   w+2,   w+3, ...     w+w
|  |  |  |  |||    |     |      | |            |
0, 2, 4, 6, ..., w*2, w*2+1, w*2+2, w*2+3, ... w*2+w*2

>
No. (ω+1)*2 = ω*2 + 2

>
No, you need to learn how ordinal arithmetic works:
>

I see: 1 + ω = ω.
>
Nevertheless the question remains where in the second row is ω
located, doesn't it?.
>

Why does it need to be there?

 
Because ω is smaller than ω2.

 
So, the bottom row is missing all the odd natural numbers. why can't it
skip all the odd multiples of ω too?
 

>
THe two set/series are of the same size even though the bottom misses
the "odd" values.

 
The two sets are of same number of elements because every elements is
doubled. But after multiplication these elements cover an interval twice
as large as before. The end is ω2. Therefore ω or its neighbours are
located in the midst of the interval.

 
Nope, unless by "midst" you mean that one is above it and the rest below it.
 
Note, your original set wasn't "equally spaced" as the ω at the end is
an infinite distance from the rest of the set, so you can't use equal
spacing logic on the second set.

Also, there is no spacing defined in sets (like image and preimage)
except N, Z, Q, R is used, which is not the case here because the
(ordered) collection of ordinal numbers and limit ordinals do are
neither existing nor defined on an equidistant scale (or row of points).