Inter-universal geometry と ABC予想 (応援スレ) 73

Inter-universal geometry と ABC予想 (応援スレ) 73 (668ﾚｽ)
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638: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/08/14(木) 10:54:05.92 ID:1dI79/KQ

つづき

U-small vs. U-large:
In this context, a set is considered "U-small" if it belongs to the universe U, and "U-large" otherwise.

Foundation for Category Theory:
This approach allows for the construction of categories with large collections of objects and morphisms, which are essential for certain areas of category theory, without encountering Russell's paradox or other foundational issues.

Alternative to ZFC:
While ZFC (Zermelo-Fraenkel set theory with the axiom of choice) is a common foundation for mathematics, MacLane's proposal provides an alternative by using the concept of a Grothendieck universe.

Key Concepts:
The use of a Grothendieck universe allows for the development of concepts like small limits and colimits within the category of U-small sets, which are fundamental in category theory.
(引用終り)

https://handwiki.org/wiki/Conglomerate_(set_theory)
Conglomerate (set theory)
From HandWiki
In mathematics, a conglomerate is a collection of classes, just as a class is a collection of sets.[1] A quasi-category is like a category except that its objects and morphisms form conglomerates instead of classes.[1] The subclasses of any class, and in particular, the collection of all classes (every class is a subclass of the class of all sets), form a conglomerate.
References
1. Adamek, Jiri; Herrlich, Horst; Strecker, George (1990). Abstract and Concrete Categories: The Joy of Cats. Dover Publications. ISBN 978-0-486-46934-8.

https://en.wikipedia.org/wiki/Conglomerate_(mathematics)
Conglomerate (mathematics)
In mathematics, in the framework of one-universe foundation for category theory,[1][2] the term conglomerate is applied to arbitrary sets as a contraposition to the distinguished sets that are elements of a Grothendieck universe.[3][4][5][6][7][8]

Definition
The most popular axiomatic set theories, Zermelo–Fraenkel set theory (ZFC), von Neumann–Bernays–Gödel set theory (NBG), and Morse–Kelley set theory (MK), admit non-conservative extensions that arise after adding a supplementary axiom of existence of a Grothendieck universe
U. An example of such an extension is the Tarski–Grothendieck set theory, where an infinite hierarchy of Grothendieck universes is postulated.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1753000052/638

643: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/08/14(木) 13:36:05.27 ID:1dI79/KQ

>>638
(引用開始)
https://handwiki.org/wiki/Conglomerate_(set_theory)
Conglomerate (set theory)
From HandWiki
In mathematics, a conglomerate is a collection of classes, just as a class is a collection of sets.[1]
A quasi-category is like a category except that its objects and morphisms form conglomerates instead of classes.[1]
The subclasses of any class, and in particular, the collection of all classes (every class is a subclass of the class of all sets), form a conglomerate.
References
1. Adamek, Jiri; Herrlich, Horst; Strecker, George (1990). Abstract and Concrete Categories: The Joy of Cats. Dover Publications. ISBN 978-0-486-46934-8.
(引用終り)

＜google訳＞
数学において、conglomerateはクラスのcollectionであり、クラスは集合のcollectionである。[1]
quasi-categoryはカテゴリに似ているが、そのオブジェクトと射がクラスではなくconglomerateを形成する点が異なる。[1]
任意のクラスのサブクラス、特にすべてのクラスの集合（すべてのクラスはすべての集合のクラスのサブクラスである）は conglomerateを形成する。
References
略
(google訳終り)

このFrom HandWiki の用語を借りれば
大きな Grothendieck Universeがあって
その中に conglomerate ＞ クラス(class) ＞ 集合(set)
という collection の大きさの違いが 存在する

この（21世紀の用語の）視点では、Grothendieck Universe は、One Universe で
Inter-universal 宇宙間 というのは、conglomerate あるいは クラス(class)
で収まるだろう

薄葉季路先生の 『集合論の宇宙　Universe と Multiverse』>>637
と比較して、望月用語”宇宙”は ちょっと 大げさ （それは Grothendieckの時代(1960年代)は それでよかったとしても）

そこらは、本当は 加藤さんあたりが 整理してほしいところです

http://rio2016.5ch.net/test/read.cgi/math/1753000052/643

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