現代数学の系譜　カントル　超限集合論

[過去ﾛｸﾞ] 現代数学の系譜　カントル　超限集合論 (1002ﾚｽ)
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216: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/10/09(水) 11:48:43.65 ID:nHmzRvjt

>>214
”ここから分出公理で
{x∈E | x: finite, x: ordered inthe sence of Neumann}
という集合がとれますがコレでいらないもが削ぎ落とされて
求めるωがとれたのでした。”

↓
E''＝E'＼N ＝ { x∈E' | x: transfinite, x: ordered in the sence of Zermelo }
という集合がとれます
コレでいらない自然数Nの元（finiteな元）が削ぎ落とされて
E'のZermelo構成の最小元として
求めるωがとれたのでした
（ここに、E'とNとは、>>211をご参照）

（参考）
https://en.wikipedia.org/wiki/Transfinite_number
Transfinite number
(抜粋)
Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were, nevertheless, not finite.
Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". However, the term "transfinite" also remains in use.

Definition
Any finite number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set (e.g., "the third man from the left" or "the twenty-seventh day of January").
When extended to transfinite numbers, these two concepts become distinct. A transfinite cardinal number is used to describe the size of an infinitely large set, while a transfinite ordinal is used to describe the location within an infinitely large set that's ordered. The most notable ordinal and cardinal numbers are, respectively:

つづく

http://rio2016.5ch.net/test/read.cgi/math/1570237031/216

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