https://www.math.columbia.edu/~woit/wordpress/?p=11709#comments woitブログ (抜粋) Peter Scholze says: April 17, 2020 at 7:15 pm PS: I just realized that maybe the following information is worth sharing. Namely, as an outsider one may wonder that the questions being discussed at length in these comments (e.g., the issue of distinct copies etc.) are very far from the extremely intricate definitions in Mochizuki’s manuscripts (his notation is famously forbidding, some of it surfaced in Taylor’s comments), and feel almost philosophical, so one might wonder that one is not looking at the heart of the matter. However, the discussions in Kyoto went along extremely similar lines, and these discussions were actually very much led, certainly initially, by Mochizuki. He first wanted to carefully explain the need for distinct copies, by way of perfections of rings, and then of the log-link, leading to discussions rather close to the one I was having with UF here. He agreed that one first has to understand these basic points before it makes sense to introduce all further layers of complexity. (I should add that we did also go through the substance of the papers, but kept getting back at how this reflects on the basic points, as we all agreed that this is the key of the matter.) 0179現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/21(火) 12:19:05.83ID:QPCMXF1q>>178 追加
www.DeepL.com/Translator(無料版) 部分修正 Peter Scholze says: April 17, 2020 at 7:15 pm PS: 以下の情報は共有する価値があるのではないかと思っています。 つまり、部外者からは、これらのコメントで長々と議論されている疑問は、以下のようなものではないかと思うでしょう。 望月の論文にある非常に複雑な定義で (e.g., the issue of distinct copies etc.) (望月の表記は禁止されていることで有名で (ここは原英文も意味不明です)、テイラーのコメントにもその一部が登場している) ほとんど哲学的な感じで、問題の核心を見ていないのではないかと疑問に思うかもしれません。 しかし、京都での議論は非常に似たような線をたどっていた、この議論は実際には、最初は確かに望月によって次のようにリードされていました。 彼はまず、the need for distinct copies, by way of perfections of rings, and then of the log-link を丁寧に説明しようとしていて、私がここでUFと議論していたのと同じような議論になった。 彼は、最初にこれらの基本的な点を理解してからでないと 複雑な他のすべての階層を導入することに意味がないとしていた。 (ここで論文の内容についても議論したが、基本的な点をどう反映させているのかということに戻ってきて、これが問題の鍵であるということに全員が同意した。) (引用終り) 0180現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/21(火) 12:28:38.49ID:QPCMXF1q>>179 補足
これをさらに要約すると 1.ショルツ先生は、これ以前は、IUT全然だめと言っていた 2.このコメントで、望月の論文にある非常に複雑な定義が問題で、自分が反例として出した例とは、すれ違いの可能性があることだけは認めた 3.このスレ違いは、京都でもあって、いまのWoitブログのDupuy氏やUF氏の指摘の議論と似た展開であったことを思い出した 4.そして、”(I should add that we did also go through the substance of the papers, but kept getting back at how this reflects on the basic points, as we all agreed that this is the key of the matter.)” と結んでいる。つまり、もうちょっと、望月論文を読まないといけない (あるいは、原文は過去形で、ちゃんと読めてないところがあって、それが、”the key of the matter”だと)
どこをどう読んだらそう読める? むしろショルツの認識は京都で議論した時点から全く変わっていないということだろ 0182現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/21(火) 15:45:34.50ID:QPCMXF1q>>181 違う ショルツ氏は思い出したんだよ 京都の議論を そして、いまのDupuyとUF氏の指摘と同じだと いう そして、>>178 (e.g., the issue of distinct copies etc.) are very far from the extremely intricate definitions in Mochizuki’s manuscripts (his notation is famously forbidding, some of it surfaced in Taylor’s comments), and feel almost philosophical, so one might wonder that one is not looking at the heart of the matter. ってこと
要するに、繰り返すが feel almost philosophical, so one might wonder that one is not looking at the heart of the matter. ってこと
で、私なりに咀嚼して解説すると これ お経みたいなものだということ
門前の小僧習わぬ教を読む 習えば、お経が読めるんだよ
戻ると 例えば ”feel almost philosophical, so one might wonder that one is not looking at the heart of the matter.” が、ある人から見れば、(わけわからん)お経でも ちゃんと、仏教の修行をした坊さんとか 分かる人が見れば 「これは ありがたい仏の教えだ」となるわけ
1.ショルツ氏も、21世紀の数学の論文において、「自分を基準にして、自分が読める論文にしろ」(それが査読の条件だ)とは言っていない 2.いままで、ショルツ氏が言ってきたことは、「自分なりに読んで、IUT Cor3.12はおかしい。矛盾があって、IUTは根本的に不成立!」と言ってきた 3.>>178の”Peter Scholze says: April 17, 2020 at 7:15 pm”は、望月氏の論文はお経で、”feel almost philosophical, so one might wonder that one is not looking at the heart of the matter.” まで 後退したってこと。つまり、上記の2の ”自分なりに読んで” の部分が成り立っていないことに、いまさらながら 気付いたってこと 4.ショルツ氏が読めない (お経のような)論文書いてどうするんだという意見は認めるとしても 上記2と3とは、決定的に違うよ 5.そして、お経じゃなく、読める論文にしてくれというのも分かる。でも、それは 論文が成立しているか不成立か とは違う議論だよ
>feel almost philosophical, so one might wonder that one is not looking at the heart of the matter. これは「(このコメントを見ている人にとって)このブログでの議論はほとんど哲学的に感じられ、問題の核心を見ていないと思うかもしれません。」 ってことだよ その後に続く部分は、「でもそうじゃないよ」っていう弁明だよ
あと、 >>179 で >(望月の表記は禁止されていることで有名で (ここは原英文も意味不明です)、テイラーのコメントにもその一部が登場している) って書いているけど、ここは「テイラーのコメントにあるように、このブログで望月の記号をそのまま書き込むことはできない」くらいの意味 該当する「テイラーのコメント」は恐らくこれのことだろう >Taylor Dupuy says: >April 14, 2020 at 5:30 pm >… >(last time I tried a double underline it didn’t work out so I’m using double prime this time, >here we need to take an analytification or formal scheme with log structure) >… https://www.math.columbia.edu/~woit/wordpress/?p=11709&cpage=2#comment-236043 0187132人目の素数さん2020/04/21(火) 16:48:59.01ID:RBkmWQJ3 JAXAの表現では137億年どまり。さて?
https://www.math.columbia.edu/~woit/wordpress/?p=11723 Not Even Wrong Why the Szpiro Conjecture is Still a Conjecture Posted on April 18, 2020 by woit (抜粋) KS says: April 20, 2020 at 11:32 am For more sociological understanding , please let me put about some curious social situation around this issue in Japan. On the official announcement of acceptance of the papers in Kyoto, two mathematicians Tamagawa and Kashiwara attended there, and Tamagawa is actually well-known expert in the anabelian geometry.But It seems likely that he can not explain about the mathematical matter of IUT in public so far. In addition , Rigid geometer Fumiharu Kato already published the book “in 2019” about IUT “treated as correct theory” for general audience amazingly, but actually it wouldn’t be sufficient to offer any insight for professional mathematicians.Furthermore , as Tamagawa is , he also has never wrote any rigorous mathematical papers about IUT as far as I know at present. These facts seems to be indeed mysterious that is , ” whoever did understand and can defense the theory sufficiently?” .
Anyway needless to say , since IUTT use too many and overlevel terminology ,so that if it cannot be expressed with crucial idea for proof , other mathematicians wouldn’t accept it at all. 0195現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/21(火) 18:09:57.50ID:QPCMXF1q>>194 なお、補足しておくが 私は、KS says に大賛成 RIMSがちゃんと、IUTが成り立つことの説明責任を果たせということ それやってほしいね 0196132人目の素数さん2020/04/21(火) 18:16:20.98ID:RBkmWQJ3 LabCuspってラブカスプっていうの?ラボカスプっていうの? Lab Cuspって分けてGoogleで翻訳するとラボカスプって出るんだけど。 0197現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/21(火) 18:29:26.20ID:QPCMXF1q>>196 さあ、LabCusp と Lab Cusp と ラボカスプ との使い分けね それは、文脈依存だから、Googleで翻訳では無理ってことじゃない? 0198現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/21(火) 18:31:57.55ID:QPCMXF1q>>195 補足
ついでに補足しておくと >>182より "(e.g., the issue of distinct copies etc.) are very far from the extremely intricate definitions in Mochizuki’s manuscripts (his notation is famously forbidding, some of it surfaced in Taylor’s comments), and feel almost philosophical, so one might wonder that one is not looking at the heart of the matter."
数学者の議論では、よく「最初に定義を確認しましょう」なんてやりますよね で、ショルツ先生、2年後になって "(e.g., the issue of distinct copies etc.) are very far from the extremely intricate definitions in Mochizuki’s manuscripts (his notation is famously forbidding, some of it surfaced in Taylor’s comments), and feel almost philosophical, so one might wonder that one is not looking at the heart of the matter." って、どういうこと? その議論の場で、定義を確認すれば良かったのに〜w 定義の確認をせずに議論して、Woitブログで、”feel almost philosophical, so one might wonder that one is not looking at the heart of the matter."とは、なんでしょうかね〜?(^^
そういうことは言わないの 弘法も筆の誤り、フィールズ賞のショルツ先生も、 定義の確認をしないで議論して、 2年後に”(e.g., the issue of distinct copies etc.) are very far from the extremely intricate definitions in Mochizuki’s manuscripts (his notation is famously forbidding, some of it surfaced in Taylor’s comments), and feel almost philosophical,” と宣うのです 正直でいい そういうことは、みなあるものですよ 0211現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/22(水) 07:57:54.59ID:s+irHIkm>>210 >>IUTが何か知らないけど応援しています >同意 >同じですw(^^;
https://www.math.columbia.edu/~woit/wordpress/?p=11723 Not Even Wrong Why the Szpiro Conjecture is Still a Conjecture Posted on April 18, 2020 by woit (抜粋) Jay Watt says: April 21, 2020 at 12:23 pm I have been following this drama for quite a while now and there’re a couple of things I just don’t get:
1. Why do people assume that the referees (if there were any) really understood the papers and should thus come out and explain it? Most of you have been refereeing papers yourself. Do you read and check every line? That’s impossible, and I’m open about this in every report I write. Even if you do think you got it all, does your judgment make the paper correct? We’re all just humans and prone to make errors. As an amusing reminder, here’s an (incomplete) list of “incomplete proofs”: https://en.wikipedia.org/wiki/List_of_incomplete_proofs. 以下略 0236132人目の素数さん2020/04/22(水) 22:06:40.22ID:M9EJX72X Iutは、どっちにせよ、先に進めるべきと思う。 いろいろな応用成果出たら疑義のある派も真剣に論駁せざるを得ないだろ。
(参考) https://www.math.columbia.edu/~woit/wordpress/?p=11723 Not Even Wrong Why the Szpiro Conjecture is Still a Conjecture Posted on April 18, 2020 by woit (抜粋) woit氏 There has been a remarkable discussion going on for the past couple weeks in the comment section of this blog posting, which gives a very clear picture of the problems with Mochizuki’s claimed proof of the Szpiro conjecture. These problems were first explained in the 2018 Scholze-Stix document Why abc is still a conjecture. In order to make this discussion more legible, and provide a form for it that can be consulted and distributed outside my blog software, I’ve put together an edited version of the discussion. I’ll update this document if the discussion continues, but it seemed to me to now be winding down. (終り)
DeepL翻訳 一部修正 このブログ記事のコメント欄では、ここ数週間、注目すべき議論が行われており、望月氏が主張しているSzpiro 憶測の証明の問題点を非常に明確に示しています。これらの問題点は、2018年のScholze-Stixのドキュメント「Why abc is still a conjecture」で最初に説明されています。 この議論をより読みやすくするために、また、私のブログソフトの外でも相談したり配布したりできる形を提供するために、この議論の編集版をまとめてみました。議論が続けばこの文書を更新しますが、私には今は風力が低下しているように見えましす。 0240現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/22(水) 23:06:43.40ID:s+irHIkm>>238 おサルさん、ご苦労さん 新左翼に反応したのかw(^^; 60年安保があって、70年安保があって
ご苦労さまですw(^^ 0241132人目の素数さん2020/04/22(水) 23:07:38.03ID:gJEljt32 Szpiro Conjectureとかって出てるけどスピロ氏4/18に亡くなってる。 0242132人目の素数さん2020/04/22(水) 23:12:55.57ID:gJEljt32 [3] Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice. PDF NEW !! (2020-04-18)
こっちの話だったんだが。
ブログで話題になった3.12ってのは上記のp173以降の事なのかな? Corollary 3.12. (Log-volume Estimates for Θ-Pilot Objects) Suppose that we are in the situation of Theorem 3.11. Write
その点、Kirti Joshi ”On Mochizuki’s idea of Anabelomorphy and its applications” ”26 Perfectoid algebraic geometry as an example of anabelomorphy 61”は注目しています ショルツ先生のPerfectoidと望月理論の関連がつけば、面白い ”[DJ] Taylor Dupuy and Kirti Joshi. Perfectoid anbelomorphy.”なんてのも、予告だけはあるw(^^; でも、ショルツ先生に間違いを指摘されて後、改訂版出すと言ってから、まだ出ていない さて、どうなることか ”バシっ”としたモノ(論文)が出れば、決定打かも
(参考) https://arxiv.org/pdf/2003.01890.pdf On Mochizuki’s idea of Anabelomorphy and its applications Kirti Joshi March 5, 2020 (抜粋) 26 Perfectoid algebraic geometry as an example of anabelomorphy 61 Now let me record the following observation which I made in the course of writing [Jos19a] and [Jos19b]. A detailed treatment of assertions of this section will be provided in [DJ] where we establish many results in parallel with classical anabelian geometry. References [DJ] Taylor Dupuy and Kirti Joshi. Perfectoid anbelomorphy. 0244現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/22(水) 23:30:06.47ID:s+irHIkm>>242 >ここ4/18で変わったのかなとか思っただけ。
まず、その話は、下記の更新情報を見ると、Corollary 3.12は変わってないと分かる (参考) http://www.kurims.kyoto-u.ac.jp/~motizuki/news-japanese.html 望月 最新情報 (抜粋) 2020年04月18日 ・(論文)修正版を更新(修正箇所のリスト): Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice.
http://www.kurims.kyoto-u.ac.jp/~motizuki/2020-04-18-iu-teich-iii-revisions.txt (修正箇所のリスト) ・Corrected misprints ("resticting" ---> "restricting", 2 instances) in Remark 1.1.1, (ii) ・Corrected a misprint ("taulological" ---> "tautological") in Remark 1.1.2, (i) (終り)
>ブログで話題になった3.12ってのは上記のp173以降の事なのかな? >Corollary 3.12. (Log-volume Estimates for Θ-Pilot Objects) Suppose that we are in the situation of Theorem 3.11. Write
(参考) https://en.wikipedia.org/wiki/Lucien_Szpiro (抜粋) Lucien Szpiro (23 December 1941 ? 18 April 2020) was a French mathematician known for his work in number theory, arithmetic geometry, and commutative algebra. He formulated Szpiro's conjecture and was a Distinguished Professor at the CUNY Graduate Center and an emeritus Director of Research [fr] at the CNRS.
Early life and education Lucien Szpiro was born on 23 December 1941 in Paris, France.[1] Szpiro attended Paris-Sud University where he earned his Ph.D. under Pierre Samuel.[1] His doctoral work was heavily influenced by the seminars of Maurice Auslander, Claude Chevalley, and Alexander Grothendieck.[1] He earned his Doctorat d'Etat (DrE) in 1971.[1]
In 1981, Szpiro formulated a conjecture (now known as Szpiro's conjecture) relating the discriminant of an elliptic curve with its conductor.[7] His conjecture inspired the abc conjecture,[8] which was later shown to be equivalent to a modified form of Szpiro's conjecture in 1988.[9] Szpiro's conjecture and its equivalent forms have been described as "the most important unsolved problem in Diophantine analysis" by Dorian Goldfeld,[10] in part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the Fermat?Catalan conjecture, and Brocard's problem.[11][12][13][14] 0246現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/23(木) 07:33:25.38ID:/do61ABJ>>244
(参考) https://www.math.columbia.edu/~woit/wordpress/?p=11723 Not Even Wrong Why the Szpiro Conjecture is Still a Conjecture Posted on April 18, 2020 by woit (抜粋) naf says: April 18, 2020 at 2:44 pm
I myself devoted most of three months way back in 2013 to going through the papers and it was a shock, on reaching Corollary 3.12, to realise that nothing is really proved. 0247132人目の素数さん2020/04/23(木) 07:41:54.30ID:pceyMXls 前に誰かが推奨してたけど、望月の出張講演資料の [11] 数論的Teichmuller理論入門 (京都大学理学部数学教室 2008年5月) が分かりやすい。そのうち、特に「談話会」がコンパクト。 http://www.kurims.kyoto-u.ac.jp/~motizuki/2008-05%20danwakai-ohp-jpg.pdf
・特定のスキーム(環)等を問題にするのではなく、そのスキームたちを統制する「抽象的組合せパターン」を主役として考える。 ・数体の異なる素点での幾何(スキーム論=環論では直接に「連絡不能な幾何」の間の連絡を実現する) ・logは環構造と両立しない。従ってスキーム(環)の幾何の中では扱えないが、Galoisと両立するため、先ほどの定理により、絶対遠アーベル幾何の枠組みで扱える ⇒Woitブログで、ScholzeがUFに何度か「The logarithm map is not a map of rings」と述べて対立した箇所。
(参考) (>>202) >>182より "(e.g., the issue of distinct copies etc.) are very far from the extremely intricate definitions in Mochizuki’s manuscripts (his notation is famously forbidding, some of it surfaced in Taylor’s comments), and feel almost philosophical, so one might wonder that one is not looking at the heart of the matter." 0249現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/23(木) 07:55:56.79ID:/do61ABJ>>247 どうも、コメントありがとう
一言追加しておくと、IUTはIVでは 下記 ”which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves. ” という大風呂敷をかましているわけ
*)ほんと蛇足だが、上記2)又は3)の状況下で、”the Szpiro Conjecture for elliptic curves”を独自に証明したら、拍手喝さいだが 上記1)の状況下では、二番煎じの別証明という評価にしかならないのです
(参考) http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV: LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki April 2020 Abstract. (抜粋) In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves. 0256現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/23(木) 11:29:55.60ID:Ayy0wuIh ”inter-universal”の用語解説があったから貼る 正直意味わからんけどw、グロタンディーク宇宙とはあんまり関係ないね。山下解説外している気がする
(参考) http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV: LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki April 2020 Abstract. (抜粋) Moreover, it is precisely these foundational issues surrounding the vertical and horizontal arrows of the log-theta-lattice that led naturally to the introduction of the term “inter-universal”.
P6 Again, the fundamental tool that makes this possible, i.e., that allows one to express constructions in the new universes in terms that makes sense in the original universe is precisely the species-theoretic formulation ? i.e., the formulation via settheoretic formulas that do not depend on particular choices invoked in particular universes ? of the constructions of interest ? cf. the discussion of Remarks 3.1.2, 3.1.3, 3.1.4, 3.1.5, 3.6.2, 3.6.3. This is the point of view that gave rise to the term “inter-universal”. At a more concrete level, this “inter-universal” contact between constructions in distant models of conventional scheme theory in the log-theta-lattice is realized by considering [the ´etale-like structures given by] the various Galois or ´etale fundamental groups that occur as [the “type of mathematical object”, i.e., species constituted by] abstract topological groups [cf. the discussion of Remark 3.6.3, (i); [IUTchI], §I3].
P7 Finally, we observe that although, in the above discussion, we concentrated on the similarities, from an “inter-universal” point of view, between the vertical and horizontal arrows of the log-theta-lattice, there is one important difference between these vertical and horizontal arrows: namely, ・ whereas the copies of the full arithmetic fundamental group - i.e., in particular, the copies of the geometric fundamental group - on either side of a vertical arrow are identified with one another, ・ in the case of a horizontal arrow, only the Galois groups of the local base fields on either side of the arrow are identified with one another - cf. the discussion of Remark 3.6.3, (ii).
P71 Remark 3.1.4. Note that because the data involved in a species is given by abstract set-theoretic formulas, the mathematical notion constituted by the species is immune to, i.e., unaffected by, extensions of the universe - i.e., such as the ascending chain V0 ∈ V1 ∈ V2 ∈ V3 ∈ ... ∈ Vn ∈ ... ∈ V that appears in the discussion preceding Definition 3.1 - in which one works. This is the sense in which we apply the term “inter-universal”. That is to say, “inter-universal geometry” allows one to relate the “geometries” that occur in distinct universes.
P81 Remark 3.6.2. (i) In the context of the theme of “coric descriptions of non-coric data” discussed in Remark 3.6.1, (ii), it is of interest to observe the significance of the use of set-theoretic formulas [cf. the discussion of Remarks 3.1.2, 3.1.3, 3.1.4, 3.1.5] to realize such descriptions. That is to say, descriptions in terms of arbitrary choices that depend on a particular model of set theory [cf. Remark 3.1.3] do not allow one to calculate in terms that make sense in one universe the operations performed in an alien universe! This is precisely the sort of situation that one encounters when one considers the vertical and horizontal arrows of the log-theta-lattice [cf. (ii) below], where distinct universes arise from the distinct scheme-theoretic basepoints on either side of such an arrow that correspond to distinct ring theories, i.e., ring theories that cannot be related to one another by means of a ring homomorphism - cf. the discussion of Remark 3.6.3 below. Indeed, it was precisely the need to understand this sort of situation that led the author to develop the “inter-universal” version of Teichm¨uller theory exposed in the present series of papers.
(参考) http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV:LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki April 2020 (抜粋) P68 Although we shall not discuss in detail here the quite difficult issue of whether or not there actually exist ZFCG-models, we remark in passing that it may be possible to justify the stance of ignoring such issues in the context of the present series of papers - at least from the point of view of establishing the validity of various “final results” that may be formulated in ZFC-models - by invoking the work of Feferman [cf. [Ffmn]]. Precise statements concerning such issues, however, lie beyond the scope of the present paper [as well as of the level of expertise of the author!]. In the following discussion, we use the phrase “set-theoretic formula” as it is conventionally used in discussions of axiomatic set theory [cf., e.g., [Drk], Chapter 1,§2], with the following proviso: In the following discussion, it should be understood that every set-theoretic formula that appears is “absolute” in the sense that its validity for a collection of sets contained in some universe V relative to the model of set theory determined by V is equivalent, for any universe W such that V ∈ W, to its validity for the same collection of sets relative to the model of set theory determined by W [cf., e.g., [Drk], Chapter 3, Definition 4.2]. 0270現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/23(木) 13:49:35.33ID:Ayy0wuIh>>269 参考に訳追加