現代数学の系譜カントル超限集合論他 3

[過去ﾛｸﾞ] 現代数学の系譜カントル超限集合論他 3 (548ﾚｽ)
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542: １３２人目の素数さん [] 2022/01/28(金) 14:34:30.29 ID:OCJDS5eR

転載しておく

Inter-universal geometry と ABC予想 (応援スレ) 64
https://rio2016.5ch.net/test/read.cgi/math/1641704497/594
594 名前：１３２人目の素数さん[sage] 投稿日：2022/01/28(金) 07:44:33.71 ID:341TuiYA
>>7 追加
>　”（スレ55 https://rio2016.5ch.net/test/read.cgi/math/1623558298/158より）
>　＜上昇列　０＜・・・＜ω　が有限列にしかなり得ない
>　ことも分からん「考えなしの素人」に数学はムリ”

反例が見つかった（下記）ｗ
下記のOrdinal arithmetic
・Addition で、... < 0'
・Multiplicationで、... < 01
・Exponentiationで、... < (0,1)
ｗｗｗ

https://en.wikipedia.org/wiki/Ordinal_arithmetic
Ordinal arithmetic

Addition
The first transfinite ordinal is ω, the set of all natural numbers. For example, the ordinal ω + ω is obtained by two copies of the natural numbers ordered in the usual fashion and the second copy completely to the right of the first. Writing 0' < 1' < 2' < ... for the second copy, ω + ω looks like
0 < 1 < 2 < 3 < ... < 0' < 1' < 2' < ...
This is different from ω because in ω only 0 does not have a direct predecessor while in ω + ω the two elements 0 and 0' do not have direct predecessors.

Multiplication
Here is ω・2:
00 < 10 < 20 < 30 < ... < 01 < 11 < 21 < 31 < ...,
which has the same order type as ω + ω.

Exponentiation
For instance, ω^2 = ω・ω using the operation of ordinal multiplication. Note that ω・ω can be defined using the set of functions from 2 = {0,1} to ω = {0,1,2,...}, ordered lexicographically with the least significant position first:
(0,0) < (1,0) < (2,0) < (3,0) < ... < (0,1) < (1,1) < (2,1) < (3,1) < ... < (0,2) < (1,2) < (2,2) < ...
Here for brevity, we have replaced the function {(0,k), (1,m)} by the ordered pair (k, m).
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1595034113/542

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