ZFC+Uのおはなし

概要

[松坂 68] における集合の濃度，順序数の定義は素朴には理解しやすい．しかし「集合全体」を「同値関係で割る」という議論は（公理的）集合論的にはアヤシイを通り越してやってはいけない議論である．本稿は公理的集合論入門と銘打って，集合論的に問題のない濃度の定義を紹介するものである．その後筆者の時間が許す限り可換代数について公理的集合論と関係する話題，すなわち選択公理と同値な命題群の紹介や Grothendieck 宇宙の話などを述べたい．本稿で引用した数々の文献について，年代が古いものが多いことからインターネット上であっても原文を見つけるのが困難な作品も少なくないが，引用したもののうち [Pea91], [Zer04], [Zer08], [Fra22a], [Sko22], [Neu23], [Neu25] については [Hei66] に英訳が収録されていることを注意しておく．本稿は東京理科大学大橋研究室において，2021 年度の卒業文集に寄稿するために書かれたものである．この文集は学部生の卒業に際し卒業論文集として企画されているものであり，筆者はほとんどの学部生とは直接の面識はないが，修士課程を修了するにあたり形式上はいったん卒業すること，また学士修了時に卒業論文を書かなかったことから，よい機会と思って楽しく本稿を書かせていただいた．本稿の内容の一部はセミナーで発表したものに基づいている．セミナーに出席していただき，コメントを頂いた浅川，榎園，大橋，金子，松本の各位に感謝する．

The definitions of set cardinality and ordinals in [Matsuzaka 68] are intuitively easy to understand. However, the argument of “taking the quotient of ‘the collection of all sets’ by an equivalence relation” goes beyond being merely dubious from the perspective of (axiomatic) set theory; it is a strictly impermissible argument. Under the title of “Introduction to Axiomatic Set Theory,” this article introduces a definition of cardinality that is free of set-theoretic problems. Subsequently, as time permits, I would like to discuss topics in commutative algebra related to axiomatic set theory—specifically, introducing a group of propositions equivalent to the Axiom of Choice and discussing Grothendieck universes.

Regarding the numerous references cited in this article, since many are quite old, it is not uncommon for the original texts to be difficult to find even on the internet. However, it should be noted that among the cited works, English translations of [Pea91], [Zer04], [Zer08], [Fra22a], [Sko22], [Neu23], and [Neu25] are included in [Hei66].

This article was written as a contribution to the Academic Year 2021 graduation anthology of the Ohashi Laboratory at the Tokyo University of Science. While this anthology is planned as a collection of graduation theses for undergraduate students—and I have no direct acquaintance with most of the undergraduates—I took this as a good opportunity to write this paper with enjoyment, given that completing a master’s program formally constitutes a “graduation” of sorts, and that I did not write a graduation thesis when completing my bachelor’s degree.

Part of the content of this article is based on a presentation given at a seminar. I would like to thank Messrs. Asakawa, Enokizono, Ohashi, Kaneko, and Matsumoto for attending the seminar and providing their comments.