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

217: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/10/09(水) 11:52:12.46 ID:nHmzRvjt

>>216
つづき

・ω (omega) is defined as the lowest transfinite ordinal number and is the order type of the natural numbers under their usual linear ordering.
・Aleph-naught, アレフ_{0}, is defined as the first transfinite cardinal number and is the cardinality of the infinite set of the natural numbers. If the axiom of choice holds, the next higher cardinal number is aleph-one, アレフ_{1}.
If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-naught. But in any case, there are no cardinals between aleph-naught and aleph-one.
The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the continuum (the set of real numbers): that is to say, aleph-one is the cardinality of the set of real numbers. (If Zermelo?Fraenkel set theory (ZFC) is consistent, then neither the continuum hypothesis nor its negation can be proven from ZFC.)
(引用終り)
以上

注：「アレフ_{0}」などは、例のアレフ記号なのだが、文字化けするのです。Alephと書くと、記号でないAlephと区別できなので、カナ書きにした(゜ロ゜;。まあ、原文読んでください（＾＾

http://rio2016.5ch.net/test/read.cgi/math/1570237031/217

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