20200403の記者会見により、望月Inter-universal Teichmuller theory (abbreviated as IUT) (下記)は、新しい局面に入りました。 査読が終り、IUTが正しいことは、99%確定です。 このスレは、IUT応援スレとします。番号は前スレ43を継いでNo.44とします。 (なお、このスレは本体IUTスレの43からの分裂スレです)
https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCller_theory Inter-universal Teichmuller theory (abbreviated as IUT) (抜粋) Contents 1 History 2 Mathematical significance (引用終り)
https://www.uvm.edu/~tdupuy/papers.html [ Taylor Dupuy's Homepage]論文集 https://www.math.arizona.edu/~kirti/ から Recent Research へ入る Kirti Joshi Recent Research論文集
<アンチIUT> https://www.math.columbia.edu/~woit/wordpress/?p=11709 (woitブログ) Not Even Wrong Latest on abc Posted on April 3, 2020 by woit https://taro-nishino.blogspot.com/2019/03/blog-post070.html(TARO-NISHINOの日記) ABC予想の壮大な証明をめぐって数学の巨人達が衝突する taro-nishino.blogspot.com/2019/03/blog-post063.html
https://ja.wikipedia.org/wiki/ABC%E4%BA%88%E6%83%B3 ABC予想 (抜粋) 出典 27 ^ “The ABC conjecture has (still) not been proved”. Persiflage (2017年12月). 2020年4月28日閲覧。 https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ The ABC conjecture has (still) not been proved Posted on December 17, 2017 by Persiflage (抜粋) Akshay Venkatesh says: December 18, 2017 at 8:45 am I couldn’t agree more. (引用終り)
アレクサンドル・グロタンディーク(Alexander Grothendieck)のアプローチは、固定された射有限群 G に対して有限 G-集合の圏を特徴付ける圏論的性質に関係している。例えば、G として ^Z と表記される群が考えられる。この群は巡回加法群 Z/nZ の逆極限である。あるいは同じことであるが、有限指数の部分群の位相に対する無限巡回群の完備化である。 すると、有限 G-集合は G が商有限巡回群を通して作用している有限集合 X であり、X の置換を与えると特定することができる。
上の例では、古典的なガロア理論との関係は、^Z を任意の有限体 F 上の代数的閉包 F の射有限ガロア群 Gal(F/F) と見なすことである。 すなわち、F を固定する F の自己同型は、 F 上の大きな有限分解体をとるように、逆極限により記述される。幾何学との関係は、原点を取り除いた複素平面内の単位円板の被覆空間として見なすことができる。
https://en.wikipedia.org/wiki/Perfectoid_space Perfectoid space In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime p.
A perfectoid field is a complete topological field K whose topology is induced by a nondiscrete valuation of rank 1, such that the Frobenius endomorphism Φ is surjective on K°/p where K° denotes the ring of power-bounded elements.
Perfectoid spaces may be used to (and were invented in order to) compare mixed characteristic situations with purely finite characteristic ones. Technical tools for making this precise are the tilting equivalence and the almost purity theorem. The notions were introduced in 2012 by Peter Scholze.[1]
Contents 1 Tilting equivalence 1.1 Almost purity theorem
https://en.wikipedia.org/wiki/P-adic_Hodge_theory p-adic Hodge theory
In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields[1] with residual characteristic p (such as Qp). The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge?Tate representation. Hodge?Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory. Further developments were inspired by properties of p-adic Galois representations arising from the etale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.
Contents 1 General classification of p-adic representations 2 Period rings and comparison isomorphisms in arithmetic geometry 3 Notes 4 References 4.1 Primary sources 4.2 Secondary sources
30 名前:geometry, an etale morphism (French: [etal]) is a morphism of schemes that is formally etale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, etale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the etale topology. The word etale is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle.[1]
応援スレ44 https://rio2016.5ch.net/test/read.cgi/math/1586655469/174-177 より 山下先生の下記 www.kurims.kyoto-u.ac.jp/~gokun/DOCUMENTS/abc2019Jul5.pdf A PROOF OF THE ABC CONJECTURE AFTER MOCHIZUKI By Go Yamashita preprint. last updated on 8/July/2019.
後ろに、Appendix A〜Cも付けてあって C.4. On the Prime Number Theorem. C.5. On the Residual Finiteness of Free Groups. とか、基本的な知識の補足もある C.6. Some Lists on Inter-universal Teichmuller Theory とかは、IUTの重要な記号の一覧ですかね
P366 A.3. Hodge-Arakelov-theoretic Comparison Theorem.で ”Note that these can be considered as a discrete analogue of the calculation of Gaussian integral is a Gaussian distribution (i.e., j → j^2) in the cartesian coordinate is a calculation in the polar coordinate ・・・” とか、望月先生の講演ネタで使っていた話の解説もあるな
Cor 3.12は P359 ”Corollary 13.13. (Log-volume Estimates for -Pilot Objects, [IUTchIII, Corollary 3.12]) We write -| log(θ)|∈ R ∪{+∞}” あと P360 ”Then we obtain -| log(q)|< -| log(θ)|” で、IUT III Cor3.12 になるけどねw(^^; (Proof.は、その直後から4ページほどある) 山下サーベイ論文は、それなりに面白いわ(^^
>>29 補足 https://en.wikipedia.org/wiki/Arakelov_theory In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions. Contents 1 Background 2 Results Background Arakelov geometry studies a scheme X over the ring of integers Z, by putting Hermitian metrics on holomorphic vector bundles over X(C), the complex points of X. This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec(Z) to be a complete variety. Results Arakelov (1974, 1975) defined an intersection theory on the arithmetic surfaces attached to smooth projective curves over number fields, with the aim of proving certain results, known in the case of function fields, in the case of number fields. Gerd Faltings (1984) extended Arakelov's work by establishing results such as a Riemann-Roch theorem, a Noether formula, a Hodge index theorem and the nonnegativity of the self-intersection of the dualizing sheaf in this context. Arakelov theory was used by Paul Vojta (1991) to give a new proof of the Mordell conjecture, and by Gerd Faltings (1991) in his proof of Serge Lang's generalization of the Mordell conjecture. Pierre Deligne (1987) developed a more general framework to define the intersection pairing defined on an arithmetic surface over the spectrum of a ring of integers by Arakelov. Arakelov's theory was generalized by Henri Gillet and Christophe Soule to higher dimensions. That is, Gillet and Soule defined an intersection pairing on an arithmetic variety. One of the main results of Gillet and Soule is the arithmetic Riemann?Roch theorem of Gillet & Soule (1992), an extension of the Grothendieck?Riemann?Roch theorem to arithmetic varieties. The arithmetic Riemann?Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties.
数論の賢人 2016年06月28日 Erica Klarreich 28歳でピーター・ショルツは数論と幾何学の間の深い繋がりを明らかにしつつある。 2010年、びっくりさせる噂が数論コミュニティに行き渡り、Jared Weinsteinに届いた。 どうやら、ボン大学の或る学生が数論における一つの不可解な証明に捧げられた288ペィジの本"Harris-Taylor"[訳注: 2001年01月にプリストン大学出版部から出版された、Michael HarrisとRichard Taylor共著の有名な本The Geometry and Cohomology of Some Simple Shimura Varietiesのこと]をたった37ペィジに再構成する論文を書いたようだ。 22歳の学生ピーター・ショルツは証明の最も複雑な部分の一つ(それは数論と幾何学の間の広範囲にわたる繋がりを扱っている)を回避する方法を発見していた。
https://arxiv.org/abs/1010.1540 The Local Langlands correspondence for $\GL_n$ over p-adic fields Peter Scholze 37 pages 7 Oct 2010 We reprove the Local Langlands Correspondence for $\GL_n$ over p-adic fields as well as the existence of ?-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor. In contrast to the proofs of the Local Langlands Correspondence given by Henniart and Harris-Taylor our proof completely by-passes the numerical Local Langlands Correspondence of Henniart. Instead, we make use of a previous result describing the inertia-invariant nearby cy
woitブログで、David Robertsは、ショルツ先生にバッサリ切られているぞw(^^; https://www.math.columbia.edu/~woit/wordpress/?p=11709&cpage=2#comments Latest on abc Posted on April 3, 2020 by woit (抜粋) David Roberts says: April 15, 2020 at 2:45 am @W gosh, thanks! Suppose you take the same argument and present it in two different languages ? one, the standard categorical language, and two, Mochizuki’s language where distinct copies of an isomorphic object are relevant for colimits and other categorical constructions. Assuming no other knowledge of what the argument actually is or how it is written, which language is more likely to conceal a subtle error in calculations or other mistake, and which language is more likely to make such mistakes easier to see? This is tricky: it depends who’s reading it. Who are you envisaging seeing mistakes? I can’t imagine (ignoring the fact this is IUT and tremendously baroque) that someone who’s had a decade of practice with their own idiosyncratic style of working would make mistakes more frequently that someone using the language of the majority, all things being equal, apart from the fact the latter person has more potential external checks and balances. This latter point I think can’t be overemphasised. Andrew Wiles was still speaking the language of his community by the time he emerged with his (first attempt at a) proof of FLT, and even engaged the help of someone else to try to check the subtle parts of the argument before that. This hasn’t happened here…
A bigger problem is the rigid commitment to definitions/structures that are explicitly admitted as being far more general than necessary (*cough* Frobenioids *cough*
48 名前:). This increases the friction for potential eyes on the IUT papers, if you’ll permit me a worrying metaphor mix.
https://www.math.columbia.edu/~woit/wordpress/?p=11709#comments Peter Scholze says: April 15, 2020 at 4:31 pm
Finally a short answer to David Roberts’ last message: I highly doubt your sentiment that the possibility of doing mistakes is not correlated with how well your language is adapted to the mathematics at hand. (引用終り) 以上 []
1.1. イントロダクション. アラケロフ幾何とは,おおまかに言って 体上の代数多様体の代りに,Z 上のスキームと 1 点 ベクトル束の代りに,1 点に計量の入ったベクトル束 を考えるというのものである. Szpiro は アラケロフ幾何について Put metrics at infinity on vector bundles and you will have a geometric intuition of compact varieties to help you. と言っている([27]). このことを簡単な場合だが,コンパクトリーマン面と Spec(Z) を対 比させることによって見てみよう.
Woitブログで Robertsに「Koshikawa」と書かれたは、 SSレポートが誤りと認めたか否かを、以前のスレに書かれてた人だよね。 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− David Roberts says: April 30, 2020 at 8:23 pm
@JE is this ‘specific mathematician’ Teruhisa Koshikawa? −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
>Peter Scholze, who everyone thinks is the greatest mathematician of this generation, says he cannot deduce 3.12 (which is the ABC conjecture, in paper #4) from 3.11 (a summary of the first 3 ABC papers) in Mochizuki’s papers.
>Koshikawa had a similar problem, and when he asked Mochizuki about it, the latter responded that the deduction is self-evident. ーーーーーーーーーーーーーーーーーーーーーーーーーーーーーーーー 13132人目の素数さん2018/06/09(土) 20:16:27.19ID:lShLrM9+ >>9 貼るならself-evidentのくだりが削除されてないやつを貼れよ無能
つまり ”JE says: May 1, 2020 at 4:57 am @Peter and David, Yes, that’s the tweet. ”です
この”Yes, that’s the tweet. ”は、下記の ”Peter Woit says: April 30, 2020 at 8:58 pm David Roberts, I assume JE is referring to this tweet https://twitter.com/MugaShohou/status/1253341200054054912” です! (deleted an unsolicited ad)
70 名前:I couldn't agree more mailto:sage [2020/05/05(火) 08:05:21.55 ID:zuve741i.net]
https://www.math.columbia.edu/~woit/wordpress/?p=11723 David Roberts says: May 3, 2020 at 5:52 pm Hi Peter, I think it worth including *in the pdf document* a mention that the discussion continued, and give a link back. You mentioned that there was content before the first comment you included, so why not also say the discussion continued? Not everyone reading that document will come to it by a link from your blog post where you mention this. I say this merely from the point of view of having a coherent scholarly record (whatever one’s view of the various positions).
”論文を読む”を、どう定義するかも問題だが それは、未定義としてw(^^ 全く読んでないってこともないでしょう ”Masaki Kashiwara, head of the team that examined the professor’s theory”(下記)を信じればね 自分が、「正しい」と確信を持てる程度には、読んだのでは?
https://www.math.columbia.edu/~woit/wordpress/?p=11723 JE says: May 1, 2020 at 4:57 am (Kashiwara was presented as head of the team that examined professor Mochizuki’s theory, see e.g. https://www.japantimes.co.jp/news/2020/04/04/national/japanese-mathematician-shinichi-mochizuki/#.XqvfxGgzaUk)
(上記URLから抜粋) https://www.japantimes.co.jp/news/2020/04/04/national/japanese-mathematician-shinichi-mochizuki/#.XrCgT6j7SUl Genius triumphs: Japanese mathematician's solution to number theory riddle validated japantimes KYODO, JIJI APR 4, 2020
“There are a number of new notions and it was hard to understand them,” Masaki Kashiwara, head of the team that examined the professor’s theory, said at a news conference.
(>>15より) https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ The ABC conjecture has (still) not been proved Posted on December 17, 2017 by Persiflage (抜粋) Akshay Venkatesh says: December 18, 2017 at 8:45 am I couldn’t agree more. (引用終り)
この発言のすぐ後に、有名なTerence Tao saysがあって、当時そちらしか見ていなかったんだw ”Terence Tao says: December 18, 2017 at 2:46 pm Thanks for this. I do not have the expertise to have an informed first-hand opinion on Mochizuki’s work, but on comparing this story with the work of Perelman and Yitang Zhang you mentioned that I am much more familiar with, one striking difference to me has been the presence of short “proof of concept” statements in the latter but not in the former, by which I mean ways in which the methods in the papers in question can be used relatively quickly to obtain new non-trivial results of interest (or even a new proof of an existing non-trivial result) in an existing field. In the case of Perelman’s work, already by the fifth page of the first paper Perelman had a novel interpretation of Ricci flow as a gradient flow which looked very promising,・・”
IUT IV で、Venkateshは、P8謝辞とP36の2箇所に出てきます(^^ (参考) 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) P8 Acknowledgements: In addition, I would like to thank Kentaro Sato for useful comments concerning the set-theoretic and foundational aspects of the present paper, as well as Vesselin Dimitrov and Akshay Venkatesh for useful comments concerning the analytic number theory aspects of the present paper.
P36 Remark 1.10.6. On the other hand, it was pointed out to the author by A. Venkatesh that in fact it is not difficult to modify the construction of these examples of abc sums given in [Mss] so as to obtain similar asymptotic estimates to those obtained in [Mss] [cf. the discussion of Remark 1.10.5, (ii)], even without taking into account the contributions at the prime 2.
76 名前:I couldn't agree less mailto:sage [2020/05/05(火) 09:42:16.12 ID:zuve741i.net]
ついでに 雪江明彦 「代数学3」を見たけど、分離閉包は扱われていなかった Coxのガロア本も、分離閉包は無かったな アルティン本も、記憶では無かった思う Van der van der Waerden ”Moderne Algebra”は、手元に無いが(読んでもいない チラ見のみ)、多分ないんじゃないかな?(^^
(参考) https://books.google.co.jp/books?id=oVfzBgAAQBAJ&printsec=frontcover&dq=Moderne+Algebra+van+der+Waerden&hl=ja&sa=X&ved=0ahUKEwjTxPTQ-ZvpAhXRBIgKHfIcC-YQ6AEINzAB#v=onepage&q=Moderne%20Algebra%20van%20der%20Waerden&f=false Moderne Algebra 著者: Bartel Eckmann L. Van der van der Waerden、 1937
K の代数的閉包 Kalg は、K の Kalg におけるすべての(代数的)分離拡大 を含むような K の唯一の分離拡大 Ksep を含む。この部分拡大は K の分離閉包(separable closure)と呼ばれる。 分離拡大の分離拡大は再び分離拡大であるので、Ksep の2次以上の有限次分離拡大は存在しない。別の言い方をすれば、K は「分離的に閉じている」代数拡大体に含まれている。これは(同型を除いて)本質的にただひとつである[5]。
一般に、K の絶対ガロワ群(英語版)は Ksep の K 上のガロワ群である[6]。
https://en.wikipedia.org/wiki/Absolute_Galois_group Absolute Galois group
In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group.
(When K is a perfect field, Ksep is the same as an algebraic closure Kalg of K. This holds e.g. for K of characteristic zero, or K a finite field.)
3.IUTを使った論文は出始めています。フェセンコ研での南出先生の強いIUTバージョン、Dupuy先生のところの Explicit Szpiro 論文、Joshi氏の”On Mochizuki’s idea of Anabelomorphy and its applications” これらの論文が出て、IUTに対する説明も、多分記者会見の玉川先生なども手伝って、分り易い
>Pure mathematics, on the other hand, seems to me a rock on which all idealism >founders: 317 is a prime, not because we think so, or because our minds are >shaped in one way rather than another, but because it is, because mathematical >reality is built that way.
Numberphile チャンネル登録者数 331万人 The abc Conjecture may have been proven by a Japanese mathematician - but what is it? More links & stuff in full description below ↓↓↓
いま、数論の大きなグループで、遠アーベルとラングランズ・プログラムがある Edward Frenkelは、ラングランズの研究者でしょ? Scholze, Peter (2013), “The Local Langlands Correspondence for GL(n) over p-adic fields”(下記)もそう
参考文献 ・Scholze, Peter (2013), “The Local Langlands Correspondence for GL(n) over p-adic fields”, Inventiones mathematicae 192 (3): 663?715, doi:10.1007/s00222-012-0420-5
171 名前:(本スレより転載) Inter-universal geometry と ABC予想 52 https://rio2016.5ch.net/test/read.cgi/math/1588702281/13 13 返信:132人目の素数さん[] 投稿日:2020/05/06(水) 08:08:54.43 ID:WFqbs44n [1/8] >査読終わりました。普通に終わりました。というよりも、いわくつきのIUTですから、念には念を入れて、丁寧に査読しました。問題なし >これをそのまま、受取れば良いのです。 https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ (I should say that whether the papers are accepted or not in a journal is pretty much irrelevant here; it’s not good enough for people to attest that they have read the argument and it is fine, someone has to be able to explain it.) (私は、論文がジャーナルに受け入れられるかどうかは、ここではほとんど関係がないと言うべきです; 人々が彼らの議論を読んで結構だと証明するのは十分ではなく、誰かがそれを説明できなければなりません。) (引用終り)
(参考) https://www.galoisrepresentations.com/about/ https://www.galoisrepresentations.com/wp-content/uploads/2019/09/cropped-brioche.jpg Persiflage Galois Representations and more! (抜粋) About A number theorist blogs (sometimes) about math. I am inspired by the blogs I have linked to, but I do not aspire to be similar to them in any particular way. I am a professor of mathematics at the University of Chicago. The (faux) anonymity is merely to confuse google, though I suspect that if you can’t work out who I am then you probably won’t get much out of my posts.
https://math.uchicago.edu/~fcale/ https://math.uchicago.edu/~fcale/img/cropped-brioche.jpg Frank Calegari (抜粋) I am a Professor at the University of Chicago. My research is in the area of algebraic number theory. I am particularly interested in the Langlands programme, especially, the notion of reciprocity linking Galois representations and motives to automorphic forms.
https://en.wikipedia.org/wiki/Frank_Calegari Frank Calegari (抜粋) Francesco Damien "Frank" Calegari is a professor of mathematics at the University of Chicago working in number theory and the Langlands program. Career Calegari won a bronze medal and a silver medal at the International Mathematical Olympiad while representing Australia in 1992 and 1993 respectively.[1] Calegari received his PhD in mathematics from the University of California, Berkeley in 2002 under the supervision of Ken Ribet.[2] (Thesis Ramification and Semistable Abelian Varieties) Calegari was a von Neumann Fellow of mathematics at the Institute for Advanced Study from 2010 to 2011.[3] He is a professor of mathematics at the University of Chicago.[4] As of 2020, Calegari is an Editor at Mathematische Zeitschrift and an Associate Editor of the Annals of Mathematics.[5][6]
Research Calegari works in algebraic number theory, including Langlands reciprocity and torsion classes in the cohomology of arithmetic groups.[4] (引用終り) 以上
https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ A third possibility is that we have (roughly) the status quo: no coup de grâce is found to kill off the approach, but at the same time the consensus remains that people can’t understand the key ideas. 3番目の可能性は、(おおよそ)現状のままであるということです。 アプローチを無効にするクーデターはありませんが、 同時に人々は主要なアイデアを理解できないというコンセンサスが残っている。
In this case, the mathematical community moves on and then, whether it be a year, a decade, or a century, when someone ultimately does prove ABC, one can go back and compare to see if (in the end) the ideas were really there after all. この場合、数学的コミュニティが進み、1年、10年、または1世紀であっても、 最終的に誰かがABCを証明したときに、戻って比較して結局そこに アイデアが本当にあったかどうかを確認できます。 (引用終り)
遠Abel曲線は双曲的曲線と同値な概念と考えられ, 実際次の結果が知られている(望月新一): k を(ある素数p に対して)p 進体Qp の有限生成拡大体に埋め込める体とする. (例えば,k が有理数体の有限生成拡大体ならばよい.) k 上の双曲的曲線Xと非特異代数多様体S に対し, k 上の定値でない射S→X の集合Hom non−const k (S,X) から k の絶対ガロア群Gk 上の連続開準同型π1(S)→π1(X) の集合 Hom open Gk (π1(S),π1(X)) をπ1(X¯k) の共役作用を法として 考えたものへの自然な写像は全単射である. この結果の証明にはp 進ホッジ理論が用いられる.
「整数論 女王」は、一説ではガウスが言ったという 正確には、下記かも ”数学は科学の女王であり、数論は数学の女王である。 Die Mathematik ist die Konigin der Wissenschaften und die Zahlentheorie ist die Konigin der Mathematik.” 「数学は理系の女王、数論は数学の女王。」DeepL訳(独語)
(参考) https://ja.wikiquote.org/wiki/%E3%82%AB%E3%83%BC%E3%83%AB%E3%83%BB%E3%83%95%E3%83%AA%E3%83%BC%E3%83%89%E3%83%AA%E3%83%92%E3%83%BB%E3%82%AC%E3%82%A6%E3%82%B9 カール・フリードリヒ・ガウス Carl Friedrich Gauss.jpg
引用 ・神は計算をされている。 ・数学は科学の女王であり、数論は数学の女王である。 Die Mathematik ist die Konigin der Wissenschaften und die Zahlentheorie ist die Konigin der Mathematik. ・狭くとも深くあれ。 Pauca sed matura.
【RIMS-1759】 Shinichi MOCHIZUKI INTER-UNIVERSAL TEICHM\"{U}LLER THEORY IV: LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS August , 2012 [RIMS1759.ps.gz] [RIMS1759.pdf]
【RIMS-1758】 Shinichi MOCHIZUKI INTER-UNIVERSAL TEICHM\"{U}LLER THEORY III: CANONICAL SPLITTINGS OF THE LOG-THETA-LATTICE August , 2012 [RIMS1758.ps.gz] [RIMS1758.pdf]
【RIMS-1757】 Shinichi MOCHIZUKI INTER-UNIVERSAL TEICHM\"{U}LLER THEORY II: HODGE-ARAKELOV-THEORETIC EVALUATION August , 2012 [RIMS1757.ps.gz] [RIMS1757.pdf]
【RIMS-1756】 Shinichi MOCHIZUKI INTER-UNIVERSAL TEICHM\"{U}LLER THEORY I: CONSTRUCTION OF HODGE THEATERS August , 2012 [RIMS1756.ps.gz] [RIMS1756.pdf]
https://en.wikipedia.org/wiki/Abc_conjecture abc conjecture Various attempts to prove the abc conjecture have been made, but none are currently accepted by the mainstream mathematical community and as of 2020, the conjecture is still largely regarded as unproven.[3][4] References 3^ Castelvecchi, Davide (3 April 2020). "Mathematical proof that rocked number theory will be published". Nature. doi:10.1038/d41586-020-00998-2. 4^ Further comment by P. Scholze at Not Even Wrong.
191 名前:acted from Mochizuki's arguments a proof of a quantitative result on abc, which could in principle give a refutation of the proof.[19] 19 ^ Vesselin, Dimitrov (14 January 2016). "Effectivity in Mochizuki's work on the abc-conjecture". arXiv:1601.03572 (引用終り)
下記 論文抜粋と訳です。誤読していると思う
https://arxiv.org/pdf/1601.03572.pdf EFFECTIVITY IN MOCHIZUKI’S WORK ON THE abc-CONJECTURE VESSELIN DIMITROV (抜粋) Abstract. This note outlines a constructive proof of a proposition in Mochizuki’s paper Arithmetic elliptic curves in general position, making a direct use of computable non-critical Belyi maps to effectively reduce the full abc-conjecture to a restricted form. Such a reduction means that an effective abc-theorem is implied by Theorem 1.10 of Mochizuki’s final IUT paper (Inter-universal Teichm¨uller theory IV: log-volume computations and set-theoretic foundations). <DeepL訳 一部修正> このノートでは、望月の論文Arithmetic Elliptic curves in general positionの命題について constructive proofを概説し、計算可能な non-critical Belyi写像を直接利用して、制限された形に対して 完全なabc-conjectureをeffectivelyに導出する。 このような導出は、望月の最終的なIUT論文の定理1.10によって、有効なabc定理が暗示されていることを意味する。 (多元的テイヒム・ミューラー理論IV:対数体積計算と集合理論的基礎)。 つづく []
P2 In his proof of the latter theorem Mochizuki makes an argument by contradiction, citing compactness of P1(Fv). This gives the appearance of ineffectivity of the claimed final result, Theorem A of [6] (the abc conjecture). The present note outlines a constructive proof, restricting for simplicity to the case (X, D) = (P1, [0] + [1] + [∞]) that is actually used in the implication “Theorem 1.10 of [6] ⇒ abc-conjecture.” This leads in principle to an explicit abc-inequality and hence, conditionally on the correctness of Mochizuki’s IUT papers, to effective Roth and Faltings theorems.
P9 5. Proof of the Theorem Following an amplification idea of Vojta for deducing the strong abcconjecture from his own conjecture with ramification for curves (see 14.4.14 and the (d) ⇒ (a) implication in Theorem 14.4.16 of [2]), the proof is executed on the Fermat curves Cn : {X^n + Y^n = Z^n} ⊂ P^2 with their distinguished Belyi maps
1.wikipedia 英文では、 ”Vesselin Dimitrov extracted from Mochizuki's arguments a proof of a quantitative result on abc, which could in principle give a refutation of the proof.[19]” 2.和文では 「ベッセリン・ディミトロフは望月の議論からABC予想に関する定量的な結果の証明を抽出した。これは原則的に証明に反駁することができた。[19]」(多分上記の訳) 3.Vesselin Dimitrovの論文中では ”Abstract. This note outlines a constructive proof of a proposition in Mochizuki’s paper Arithmetic elliptic curves in general position, making a direct use of computable non-critical Belyi maps to effectively reduce the full abc-conjecture to a restricted form. Such a reduction means that an effective abc-theorem is implied by Theorem 1.10 of Mochizuki’s final IUT paper (Inter-universal Teichm¨uller theory IV: log-volume computations and set-theoretic foundations).” 「<DeepL訳 一部修正> このノートでは、望月の論文Arithmetic Elliptic curves in general positionの命題について constructive proofを概説し、計算可能な non-critical Belyi写像を直接利用して、制限された形に対して 完全なabc-conjectureをeffectivelyに導出する。 このような導出は、望月の最終的なIUT論文の定理1.10によって、有効なabc定理が暗示されていることを意味する。 (多元的テイヒム・ミューラー理論IV:対数体積計算と集合理論的基礎)。」 てことで、”定理1.10の最終版では、ちゃんと 有効なabc定理が導ける可能性があるよ”と読みました
つまり、wikipedia 英文の本文の ”which could in principle give a refutation of the proof.[19]”とは、意味が真逆と思う(^^;
>>168 (引用開始) 【RIMS-1758】 Shinichi MOCHIZUKI INTER-UNIVERSAL TEICHM\"{U}LLER THEORY III: CANONICAL SPLITTINGS OF THE LOG-THETA-LATTICE August , 2012 [RIMS1758.ps.gz] [RIMS1758.pdf] (引用終り)
IUT III Corollary 3.12の証明 2012年版 を、スナップショットしておく P113〜121まで、約8頁ある
P113 Corollary 3.12. (Log-volume Estimates for Θ-Pilot Objects) Suppose that we are in the situation of Theorem 3.11. Write ? |log(Θ)| ∈ R ∪{+∞} for the procession-normalized mono-analytic log-volume
Proof. Suppose that we are in the situation of Theorem 3.11. We begin by reviewing precisely what is achieved by the various portions of Theorem 3.11 and, indeed, by the theory developed thus far in the present series of papers. This review leads naturally to an interpretation of the theory that gives rise to the inequality asserted in the statement of Corollary 3.12. For ease of reference, we divide our discussion into steps, as follows.
P121 Put another way, one must contend with the indeterminacy arising from the fact that, unlike the case with the global Frobenioids “F◎_MOD”, “F◎R_MOD”, objects of the various local Frobenioids that arise admit endomorphisms which are not automorphisms. This indeterminacy has the effect of rendering meaningless any attempt to perform a precise log-volume computation as in (xi).
ある開被覆が存在し、この微分形式の対数微分としての局所表現が存在する(通常の微分作用素 d/dz の中の外微分 d を少し変形する)。ω が整数の留数の単純極を持つだけであることに注意する。 高次元の複素多様体では、ポアンカレ留数(英語版)(Poincare residue)は、極に沿った対数的微分形式の振る舞いを記述することに使われる。
目次 1 正則対数複体 1.1 高次元の例 1.2 ホッジ理論
ホッジ理論 正則対数複体は、複素代数多様体のホッジ理論への適用することが可能である。X を複素代数多様体、 j:X\hookrightarrow Y} j:X\hookrightarrow Y} を良いコンパクト化とする。このことは Y がコンパクト代数多様体で、D = Y ? X が Y 上の単純な横断的交叉をもつ因子であることを意味する。層の複体の自然な包含写像 Ω*_{Y}(log D)→ j*Ω*_{X} は、擬同型であることがわかる。 古典的には、たとえば、楕円函数の理論の中では、対数的微分形式は第一種微分形式(英語版)(differentials of the first kind)の補完物と考えられてきた。 対数的微分形式は、第二種微分形式と呼ばれることもある(不幸にも、第三種微分形式との間に不整合がある)。古典論は、現在では、ホッジ理論の一面として取り込まれている。 たとえば、あるリーマン面 S に対し、第一種微分形式は、H1(S) の項 H1,0 として考えられている。ドルボー同型により層コホモロジー群 H0(S,Ω) として解釈すると、これらの定義は同義と考えられる定義である。 0 が S 上の正則函数 の層であるとき、 H1(S,O) と解釈できるように、H1(S) の中の H1,0 直和を、対数的微分形式のベクトル空間として、より具体的にみなすことができる。
>>181 In algebraic geometry, a log structure provides an abstract context to study semistable schemes, and in particular the notion of logarithmic differential form and the related Hodge-theoretic concepts.
(原文) https://inference-review.com/article/a-crisis-of-identification A Crisis of Identification David Michael Roberts (抜粋) 参考文献 3. PS (Peter Scholze), December 21, 2017, comment on “The ABC Conjecture Has (Still) Not Been Proved,” Persiflage, December 17, 2017. ? https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/#comment-4619
PS says: December 21, 2017 at 3:28 pm Thanks for the wonderful post! I agree with everything that was said.
One small thing I would like to add is that most accounts indicate that no experts have been able to point to a place where the proof would fail. This is in fact not the case; since shortly after the papers were out I am pointing out that I am entirely unable to follow the logic after Figure 3.8 in the proof of Corollary 3.12 of Inter-universal Teichmuller theory part III: “If one interprets the above discussion in terms of the notation introduced in the statement of Corollary 3.12, one concludes [the main inequality].” Note that this proof is in fact the *only* proof in parts II and III that is longer than a few lines which essentially say “This follows from the definitions”. Those proofs, by the way, are completely sound, very little seems to happen in those two papers (to me). Since then, I have kept asking other experts about this step, and so far did not get any helpful explanation.
In fact, over the years more people came to the same conclusion; from everybody outside the immediate vicinity of Mochizuki, I heard that they did not understand that step either. The ones who do claim to understand the proof are unwilling to acknowledge that more must be said there; in particular, no more details are given in any survey, including Yamashita’s, or any lectures given on the subject (as far as they are publicly documented). [I did hear that in fact all of parts II and III should be regarded as an explanation of this step, and so if I am unable to follow it, I should read this more carefully… For this reason I did wait for several years for someone to give a better (or any) explanation before speaking out publicly.] One final point: I get very annoyed by all references to computer-verification (that came up not on this blog, but elsewhere on the internet in discussions of Mochizuki’s work). The computer will not be able to make sense of this step either. The comparison to the Kepler conjecture, say, is entirely misguided: In that case, the general strategy was clear, but it was unclear whether every single case had been taken care of. Here, there is no case at all, just the claim “And now the result follows”. (引用終り) 以上
https://www.uvm.edu/~tdupuy/papers.html [ Taylor Dupuy's Homepage] 3.Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12, (with A. Hilado) []
Acknowledgementsに、Kiran Kedlaya、Emmanuel Lepage、Chung Pang Mok、Thomas Scanlon 達の名前が挙がっている ”Preparatory Center for Research in Next-Generation Geometry located at RIMS”も、挙がっているね
https://www.uvm.edu/~tdupuy/papers.html [ Taylor Dupuy's Homepage]論文集 https://arxiv.org/pdf/2004.13108.pdf 3.Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12, (with A. Hilado) Date: April 30, 2020. (抜粋) P4 Acknowledgements. This article is very much indebted to many previous expositions of IUT including (but not limited to) [Fes15, Hos18, Ked15, Hos15, Sti15, Mok15, Moc17, Yam17, Hos17, Tan18, SS17]. The first author also greatly benefitted from conversations with many other mathematicians and would especially like to thank Yuichiro Hoshi for helpful discussions regarding Kummer theory and his patience during discussions of the theta link and Mochizuki’s comparison; Kirti Joshi for discussions on deformation theory in the context of IUT; Kiran Kedlaya for productive discussions on Frobenioids, tempered fundamental groups, and global aspects of IUT; Emmanuel Lepage for helpful discussions on the p-adic logarithm, initial theta data, aut holomorphic spaces, the log-kummer correspondence, theta functions and their functional equations, tempered fundamental groups, log-structures, cyclotomic synchronization, reconstruction of fundamental groups, reconstruction of decomposition groups, the ”multiradial representation of the theta pilot object”, the third indeterminacy, the second indeterminacy, discussions on Hodge Theaters, labels, and kappa coric functions, and discussions on local class field theory;
Shinichi Mochizuki for his patience in clarifying many aspects of his theory ? these include discussions regarding the relationship between IUT and Hodge Arakelov theory especially the role of ”global multiplicative subspaces” in IUT, discussions on technical hypotheses in initial theta data; discussions on Theorem 3.11 and ”(abc)-modules”, discussions on mono-theta environments and the interior and exterior cyclotomes, discussions of the behavior of various objects with respect to automorphisms and providing comments on treatment of log-links and the use of polyisomorphisms, discussions on indeterminacies and the multiradial representation, discussions of the theta link, discussions on various incarnations of Arakelov Divisors, discussions on cyclotomic synchronization; Chung Pang Mok for productive discussions on the p-adic logarithm, anabelian evaluation, indeterminacies, the theta link, and hodge theaters; Thomas Scanlon for discussions regarding interpretations and infinitary logic as applied to IUT and anabelian geometry. We apologize if we have forgotten anybody.
The research discussed in the present paper profited enormously from the generous support of the International Joint Usage/Research Center (iJU/RC) located at Kyoto Universities Research Institute for Mathematical Sciences (RIMS) as well as the Preparatory Center for Research in Next-Generation Geometry located at RIMS. (引用終り) 以上
V(F)non: the set of nonarchimedean places of F V(F)arc: the set of archimedean places of F V(F) def = V(F)non ∪ V(F)arc
§1 Introduction Main theorem of IUTch: There exist “multiradial representations”? i.e., description up to mild indeterminacies in terms that make sense from the point of view of an alien ring structure ? of the following data:
⇒ As an application, we obtain a diophantine inequality.
P7 Theorem (ABC Conjecture for number fields) Note: We do not know the constant “C(d, ?)” explicitly. For instance, it is hard to compute noncritical Belyi maps explicitly!
P8 Goal of this joint work: Under certain conditions, we prove (*) directly [i.e., without applying the theory of noncritical Belyi maps] to compute the constant “C(d, ?)” explicitly. Technical Difficulties of Explicit Computations (i) We cannot use the compactness of “K” at the place 2 ⇒ We develop the theory of
242 名前:Letale theta functions so that it works at the place 2 (ii) We cannot use the compactness of “K” at the place ∞ ⇒ By restricting our attention to “special” number fields, we “bound” the archimedean portion of the “height” of the elliptic curve “Eλ”
P9 §2 Theta Functions
P10 Now we have the following sequence of log tempered coverings:
P11 ? Next, we recall the def’n of the theta function Θ¨ .
P14 We want to develop the theory of Θ functions in the case of p = 2. ⇒ In this work, instead of “2-torsion points”, we consider 6-torsion points of X(K)!
P15 §3 Heights First, we recall the notion of the Weil height of an algebraic number.
P21 §5 Expected Main Results Expected Theorem (Effective ABC for mono-complex number fields)
Expected Corollary (Application to Fermat’s Last Theorem) ∃ explicitly computable n0 ∈ Z?3 s.t. if n ? n0, then no triple (x, y, z) of positive integers satisfies x^n + y^n = z^n (引用終り) 以上 []
(参考) www.kurims.kyoto-u.ac.jp/~motizuki/news-japanese.html 望月 最新情報 2012年08月30日 ・(論文)新論文を掲載: Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation. Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice. Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations.
メモ:長い証明リスト https://en.wikipedia.org/wiki/Mathematical_proof Mathematical proof See also ・List of incomplete proofs ・List of long proofs
https://en.wikipedia.org/wiki/List_of_long_mathematical_proofs List of long mathematical proofs (抜粋) Contents 1 Long proofs 2 Long computer calculations 3 Long proofs in mathematical logic
Long proofs The length of unusually long proofs has increased with time. As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof. ・1799 The Abel?Ruffini theorem was nearly proved by Paolo Ruffini, but his proof, spanning 500 pages, was mostly ignored and later, in 1824, Niels Henrik Abel published a proof that required just six pages. ・1963 Odd order theorem by Feit and Thompson was 255 pages long, which at the time was over 10 times as long as what had previously been considered a long paper in group theory. ・1964 Resolution of singularities Hironaka's original proof was 216 pages long; it has since been simplified considerably down to about 10 or 20 pages. ・2000 Lafforgue's theorem on the Langlands conjecture for the general linear group over function fields. Laurent Lafforgue's proof of this was about 600 pages long, not counting many pages of background results. ・2003 Poincare conjecture, Geometrization theorem, Geometrization conjecture. Perelman's original proofs of the Poincare conjecture and the Geometrization conjecture were not lengthy, but were rather sketchy. Several other mathematicians have published proofs with the details filled in, which come to several hundred pages. ・2004 Classification of finite simple groups. The proof of this is spread out over hundreds of journal articles which makes it hard to estimate its total length, which is probably around 10000 to 20000 pages.
4章=IUT IVだね、きっと そして、P67のSection 3の下記引用部分だね (参考) 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 P67 Section 3: Inter-universal Formalism: the Language of Species In the following discussion, we shall work with various models - consisting of “sets” and a relation “∈” - of the standard ZFC axioms of axiomatic set theory [i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].
つづき The various ZFC-models that we work with may be thought of as [but are not restricted to be!] the ZFC-models determined by various universes that are sets relative to some ambient ZFC-model which, in addition to the standard axioms of ZFC set theory, satisfies the following existence axiom [attributed to the “Grothendieck school” ? cf. the discussion of [McLn], p. 193]: (†G) Given any set x, there exists a universe V such that x ∈ V . We shall refer to a ZFC-model that also satisfies this additional axiom of the Grothendieck school as a ZFCG-model. This existence axiom (†G) implies, in particular, that: (引用終り)
1.ZFC公理系の公理の数が9個ではなく、 今の論文では、”[i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].” だから、普通には、ZFが9個でしょ? それは、下記のZF wikipedia の9個と合う 2.かつ、” - cf., e.g., [Drk], Chapter 1, §3]”と書いてあるから、” [Drk], Chapter 1, §3]”をチェックしての発言なのかな? 自分は[Drk]をチェックする気が無いけどw 3.だから、ZFが9個で、ZFCなら10個って話かな? 元の2012年版の記憶で書いているのかな? 意味不明ですね(^^;
(参考) https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory Zermelo?Fraenkel set theory
Contents 1 History 2 Axioms 2.1 1. Axiom of extensionality 2.2 2. Axiom of regularity (also called the axiom of foundation) 2.3 3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension) 2.4 4. Axiom of pairing 2.5 5. Axiom of union 2.6 6. Axiom schema of replacement 2.7 7. Axiom of infinity 2.8 8. Axiom of power set 2.9 9. Well-ordering theorem 3 Motivation via the cumulative hierarchy
1.>>232より 望月 IUT IV ”[i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].” つまり、”nine axioms”であること、”of Zermelo-Fraenkel”(ZFであってZFCではない)ことを 確認願います。 2.よって、望月氏の記述は”式ではありません”! 3.>>233 より 「は、”個”の定義の話だと思う」と書いた ”axiom”を1個と数えれば、下記の2.1 〜2.9 9個 (但し、”9. Well-ordering theorem”は ”axiom”でないとすれば、8”axiom”+1”Well-ordering theorem”=9 という計算もありだろう) QED(^^;
(参考>>232より再録) https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory Zermelo-Fraenkel set theory
Contents 1 History 2 Axioms 2.1 1. Axiom of extensionality 2.2 2. Axiom of regularity (also called the axiom of foundation) 2.3 3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension) 2.4 4. Axiom of pairing 2.5 5. Axiom of union 2.6 6. Axiom schema of replacement 2.7 7. Axiom of infinity 2.8 8. Axiom of power set 2.9 9. Well-ordering theorem 3 Motivation via the cumulative hierarchy (引用終り)
ほいよ >>236 あなたも、ここで論陣を張りたければ、まずは事実を確認してくださいね まずは、望月氏 IUT IVが引用している >>230の”cf., e.g., [Drk], Chapter 1, §3”を見ましょうね (P85 Bibliography [Drk] F. R. Drake, Set Theory: an Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, North-Holland (1974). です)
F. R. Drake, Set Theory: an Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, North-Holland (1974). 静岡大学附属図書館と 新潟大学附属図書館とがヒットしますね(^^;
アマゾン/Set-Theory-Introduction-Foundations-Mathematics/dp/0720422795 Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics) (英語) ハードカバー ? 1974/10/1 F. R. Drake (著)
Set theory : an introduction to large cardinals | 静岡大学附属図書館 ...opac.lib.shizuoka.ac.jp ? opacid Google Books. ブックマーク済み. Set theory : an introduction to large cardinals ... North-Holland, 1974; 形態: xii, 351 p. ; 23 cm; 著者名: Drake, F. R. (Frank ... シリーズ名: Studies in logic and the foundations of mathematics ; v. 76 ... 410.8/179/76.
Set theory : an introduction to large cardinals | 新潟大学附属図書館 ...opac.lib.niigata-u.ac.jp ? opc ? recordID ? catalog.bib 9780720422795 [0720422795] (North-Holland) CiNii Books Webcat Plus Google Books; シリーズ名: Studies in logic and the foundations of mathematics ; v. 76 ...
>>239より 望月氏は ”[i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].” これ、”- cf., e.g., [Drk], Chapter 1, §”を、「[Drk]には、こう書いてあるけれども」と、軽く読めば良いんじゃない?(^^
”Well-ordering theorem”は、最初 Zermeloは定理だと考えていたのですね で、下記のように、1階述語論理では、選択公理や Zorn's Lemmaと equivalentだと (ここまでは 学部生でも常識でしょうね) しかし、2階述語論理では、strictly stronger than the axiom of choice だと なるほどね(^^;
(参考) https://en.wikipedia.org/wiki/Well-ordering_theorem Well-ordering theorem (抜粋) "Zermelo's theorem" redirects here. For Zermelo's theorem in game theory, see Zermelo's theorem (game theory). Not to be confused with Well-ordering principle.
In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents).[1][2]
History
It turned out, though, that the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with the Zermelo?Fraenkel axioms is sufficient to prove the other, in first order logic (the same applies to Zorn's Lemma). In second order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem.[7]
There is a well-known joke about the three statements, and their relative amenability to intuition:
The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?[8]
>>236 >https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory >Zermelo-Fraenkel set theory > 2.3 3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension) (引用終り)
追加 これ、現代では大分見直しされているようですね(^^; https://en.wikipedia.org/wiki/Axiom_schema_of_specification Axiom schema of specification (抜粋) In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.
Because restricting comprehension avoided Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Godel considered it the most important axiom of set theory.
Relation to the axiom schema of replacement The axiom schema of separation can almost be derived from the axiom schema of replacement. For this reason, the axiom schema of specification is often left out of modern lists of the Zermelo?Fraenkel axioms. However, it's still important for historical considerations, and for comparison with alternative axiomatizations of set theory, as can be seen for example in the following sections. Unrestricted comprehension Accepting only the axiom schema of specification was the beginning of axiomatic set theory. Most of the other Zermelo?Fraenkel axioms (but not the axiom of extensionality, the axiom of regularity, or the axiom of choice) then became necessary to make up for some of what was lost by changing the axiom schema of comprehension to the axiom schema of specification ? each of these axioms states that a certain set exists, and defines that set by giving a predicate for its members to satisfy, i.e. it is a special case of the axiom schema of comprehension.
どうも 全く同意 基礎論のプロ数学者の専門家がいうならともかくも ド素人がイチャモン付けるなら せめて 原典の>>241 F. R. Drake, Set Theory: an Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, North-Holland (1974). くらいは、当たってからにしてくれよ、おい って話ですねw(^^;
(参考) 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 P67 In the present §3, we develop ? albeit from an extremely naive/non-expert point of view, relative to the theory of foundations! ? the language of species. Roughly speaking, a “species” is a “type of mathematical object”, such as a “group”, a “ring”, a “scheme”, etc.
https://ncatlab.org/nlab/show/species species (抜粋) Contents 1. Idea 2. Definition 1-categorical 2-categorical (∞,1)-categorical Operations on species Sum Cauchy product Hadamard product Dirichlet product Composition product 3. In Homotopy Type Theory Operations on species Coproduct Hadamard product Cauchy product Composition 4. Properties Cardinality 5. Variants
1. Idea A (combinatorial) species is a presheaf or higher categorical presheaf on the groupoid core(FinSet), the permutation groupoid. A species is a symmetric sequence by another name. Meaning: they are categorically equivalent notions.
https://en.wikipedia.org/wiki/Combinatorial_species Combinatorial species (抜粋) In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for analysing discrete structures in terms of generating functions. Examples of discrete structures are (finite) graphs, permutations, trees, and so on; each of these has an associated generating function which counts how many structures there are of a certain size.
Category theory provides a useful language for the concepts that arise here, but it is not necessary to understand categories before being able to work with species.
The category of species is equivalent to the category of symmetric sequences in finite sets.[1]
.Category theory provides a useful language for the concepts that arise here, but it is not necessary to understand categories before being able to work with species. (引用終り) 以上
.Category theory provides a useful language for the concepts that arise here, but it is not necessary to understand categories before being able to work with species. ↓ 1行削除 なんか同じ行で、ダブっているね(^^;
(参考) https://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf The Foundations of Mathematics Kenneth Kunen PDF 2007/10/29 - c 2005,2006,2007 Kenneth Kunen. Kenneth Kunen
P10 I.2 The Axioms Axiom 0. Set Existence. ∃x(x = x) Axiom 1. Extensionality. ∀z(z ∈ x ←→ z ∈ y) → x = y Axiom 2. Foundation. ∃y(y ∈ x) → ∃y(y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y)) Axiom 3. Comprehension Scheme. For each formula, φ, without y free, ∃y∀x(x ∈ y ←→ x ∈ z ∧ φ(x)) Axiom 4. Pairing. ∃z(x ∈ z ∧ y ∈ z) Axiom 5. Union. ∃A∀Y ∀x(x ∈ Y ∧ Y ∈ F → x ∈ A) Axiom 6. Replacement Scheme. For each formula, φ, without B free, ∀x ∈ A∃!y φ(x, y) → ∃B ∀x ∈ A∃y ∈ B φ(x, y) Axiom 7. Infinity. ∃x({} ∈ x ∧ ∀y ∈ x(S(y) ∈ x))注:{}は空集合 Axiom 8. Power Set. ∃y∀z(z ⊆ x → z ∈ y) Axiom 9. Choice. {} not∈ F ∧ ∀x ∈ F ∀y ∈ F(x ≠ y → x ∩ y = {}) → ∃C ∀x ∈ F(SING(C ∩ x)) 注:{}は空集合
キューネン先生の”SET THEORY An Introduction to Independence Proofs”(1999)(下記) では ZFC is the system of Axioms 0-9. ZF consists of Axioms 0-8, として、Axiom 0 を含めているね
同意だ 望月論文 IUT IVでは、ちゃんと引用の教科書 [Drk], Chapter 1, §3]. [Drk] F. R. Drake, Set Theory: an Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, North-Holland (1974). これを見ないで、無限だとかアホをいうやつが、バカだってことよ(^^
(>>230-232より 望月 IUT IV) ”[i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].” 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 P67 Section 3: Inter-universal Formalism: the Language of Species In the following discussion, we shall work with various models - consisting of “sets” and a relation “∈” - of the standard ZFC axioms of axiomatic set theory [i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].
P85 Bibliography [Drk] F. R. Drake, Set Theory: an Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, North-Holland (1974).
************************************************* The last part of (IUTT-IV) explores the use of different models of ZFC set theory in order to more fully develop inter-universal Teichmuller theory (this part is not needed for the applications to the abc conjecture). There appears to be an inaccuracy in a remark in Section 3, page 43 of that paper regarding the conservative nature of the extension of ZFC by the addition of the Grothendieck universe axiom; see this blog comment. However, this remark was purely for motivational purposes and does not impact the proof of the abc conjecture. ************************************************
1.「ZFCの9個の公理」は、当時の記述がどうだかしらないが、2020年版では (>>230-232より 望月 IUT IV) ”[i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].” ってなっているので、 [Drk], Chapter 1, §3].からの引用というスタイルだから、無問題 2.「ZFCGがZFCの保存的拡大」も、当時どうだったか知らないが 2020年版では、消されているので、そこをほじくっても、何にもでないよね
>>314 補足 (引用開始) >キューネンの下記では、「ZFC = Axioms 1-9. ZF = Axioms 1-8.」と説明しているな!ww(^^; キューネン先生の”SET THEORY An Introduction to Independence Proofs”(1999)(下記) では ZFC is the system of Axioms 0-9. ZF consists of Axioms 0-8, として、Axiom 0 を含めているね (引用終り)
私としては、数学の解き方を学ぶために本を買うなら、タオ先生の "Solving Mathematical Problems: A Personal Perspective" アマゾン/exec/obidos/ASIN/B00BEAYB32/hatena-blog-22/ という本の方がいいかと思います。この本は「難しく」はないので「楽しく」読み進めることができ、どんな人にもオススメです。
タオ先生のブログ: 245A: Problem solving strategies | What's new https://terrytao.wordpress.com/2010/10/21/245a-problem-solving-strategies/
IUT-4で以下を修正しており、指摘で修正が必要であったのは、この箇所だよ。 concerning the “conservative extensionality” of ZFCG relative to ZFC, i.e., roughly speaking, that“any proposition that may be formulated in a ZFC-model and, moreover, holds in a ZFCG-model infact holds in the original ZFC-model”
>ZFCの9個の公理 のIUT-4の記載部分は、投稿版も最終版も、以下記載だったが、修正されてないので此処は問題でないよ。 ”[i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].” だいたい [Drk], Chapter 1, §3].の出典元の引用で、些末なことで、タオが此処を問題しないと思うよ。 だから確認したら、此処とは違うところの指摘だよ。
置換公理から ”On the basis of Axioms 1,3,4,5, define ⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) by” と書かれていて、置換公理って、こうやって使うのか〜w、と感心したのです で、まあ ⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) ・・・ 達を、公理 Axioms 1,3,4,5 & Axiom 6(Replacement Scheme) とか(勿論 他の公理も)使って、通常の集合論の記号や用語を組立て さらには、定理を作って・・と出来るのです(多分ねw)
(参考) https://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf The Foundations of Mathematics Kenneth Kunen PDF 2007/10/29 - c 2005,2006,2007 Kenneth Kunen. Kenneth Kunen (抜粋)
P10 Axiom 6. Replacement Scheme. For each formula, φ, without B free, ∀x ∈ A∃!y φ(x, y) → ∃B ∀x ∈ A∃y ∈ B φ(x, y)
P11 Axiom 6. Replacement Scheme. For each formula, ψ, without B free, ∀x ∈ A∃!y ψ(x, y) → ∃B ∀x ∈ A∃y ∈ B ψ(x, y) The rest of the axioms are a little easier to state using some defined notions. On the basis of Axioms 1,3,4,5, define ⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) by: x ⊆ y ⇔ ∀z(z ∈ x → z ∈ y) x = Φ ⇔ ∀z(z not∈ x) y = S(x) ⇔ ∀z(z ∈ y ←→ z ∈ x ∨ z = x) w = x ∩ y ⇔ ∀z(z ∈ w ←→ z ∈ x ∧ z ∈ y) SING(x) ⇔ ∃y ∈ x ∀z ∈ x(z = y) (引用終り)
1.”⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) ・・・ 達を、公理 Axioms 1,3,4,5 & Axiom 6(Replacement Scheme) とか(勿論 他の公理も)使って、通常の集合論の記号や用語を組立て” って話なんだよね 2.で、⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) ・・・ これらは、全部公理じゃない! 3.なぜならば、これらを導く論理式 ψってのは、それぞれ 他の公理から導かれるものだから だから、くどいが 個々の”これらを導く論理式 ψ”達ってさ、これらは公理じゃないよね?w (∵ 他の公理から導かれるものは、公理ではない) 4.だったら、置換公理”Axiom 6. Replacement Scheme. For each formula, ψ, without B free,”って、やっぱり1つと数えるべきだと思うぜ 論理式 ψに具体的に、他の公理達(含む Axiom 6)から導かれた ”formula ψ”を適用したものは、これは もう公理として扱うべきでない! それが、公理主義の真っ当な思想でしょ?! 5.だから、繰返すが 置換公理”Axiom 6. Replacement Scheme. For each formula, ψ, without B free,”は、公理としてはあくまで、具体的な ”formula ψ”を当てはめる前の状態のことで そう考えて、1つなんだよね!
https://en.wikipedia.org/wiki/Class_(set_theory) Class (set theory) (抜粋) Examples The collection of all algebraic structures of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. I n category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.
Classes in formal set theories Another approach is taken by the von Neumann?Bernays?Godel axioms (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. However, the class existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG to be a conservative extension of ZF. (引用終り)
>>383 >However, the class existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. >This causes NBG to be a conservative extension of ZF.
https://en.wikipedia.org/wiki/Conservative_extension Conservative extension (抜粋) In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is not conservative, and can prove more theorems than the original. Contents 1 Examples 2 Model-theoretic conservative extension (引用終り)
要は(^^ conservative extension:often convenient for proving theorems, but proves no new theorems about the language of the original theory non-conservative extension:can prove more theorems than the original ってこと ZFCに対し、NBGは conservativeで、ZFCGは non-conservative です
(参考:本スレより転載) Inter-universal geometry と ABC予想 52 https://rio2016.5ch.net/test/read.cgi/math/1588702281/606 (抜粋) 606 名前:132人目の素数さん[] 投稿日:2020/05/09(土) 17:57:38.91 ID:/BYRDNlz >案の定ZFCGがZFCの保存的拡大だのZFCの9個の公理だの間違えまくってる と書いてガゼで主張しただけだよ。 IUT-4で以下を修正しており、指摘で修正が必要であったのは、この箇所だよ。 concerning the “conservative extensionality” of ZFCG relative to ZFC, i.e., roughly speaking, that“any proposition that may be formulated in a ZFC-model and, moreover, holds in a ZFCG-model infact holds in the original ZFC-model” (引用終り)
>>383 まず訂正 I n category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category. ↓ In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.
この手法を任意の自然数 n で拡張し、n-圏(n-category、n 次圏)を定義することができる。さらに順序数 ω に対する ω-category と呼ばれる高次圏もある。 空間を圏で表す (O, ?) が順序集合のとき、これを次のような圏 CO と同一視することができる:obj(CO) = O とし、p, q ∈ O = obj(CO) について p ? q のとき、およびそのときに限り p から q への射がただ 1 つ存在する、として CO における射を定める。 ここで順序関係の推移律が射の合成に、反射律が恒等射に対応している。特に位相空間 X に対してその開集合系 O(X) を圏と見なすことができる。
G が群のとき、対象 Y ただ 1 つからなり、Hom (Y, Y) ≡ G であるような圏を G と同一視することができる。また、位相空間の基本亜群や「被覆」のホロノミー亜群など、様々な亜群による幾何学的な情報の定式化が得られている。
圏論 圏論に歴史的につながる宇宙への別のアプローチの方法がある。これはグロタンディーク宇宙と呼ばれる。大まかに言えば、グロタンディーク宇宙とは集合論の通常実行されるすべての操作を内部にもつ集合である。 例えば、グロタンディーク宇宙 U における2つの集合の和集合も U の内部にある。同様に、共通部分、順序対、冪集合などもまた U の内部にある。これは上記の上部構造に類似している。 グロタンディーク宇宙の利点は、それが実際の集合であって固有類ではないことである。グロタンディーク宇宙の難点は、厳密さを欲するなら、グロタンディーク宇宙を捨てなければならないことである。
最も一般的なグロタンディーク宇宙 U の用途はすべての集合の圏を U で置き換えるものである。S ∈U のとき、U-large でないなら、集合S は U-small となる。すべての U-small 集合の圏 U-Set は、すべての U-small の集合を対象として、それらの集合の間のすべての関数を射としてもつ。 対象の集合と射の集合の両方共集合であり、このことが固有類を用いることなく "すべての" 集合の圏を議論することを可能にしている。すると、この新しい圏の観点から別の圏の定義が可能になる。例えば、すべての U-small 圏の圏は宇宙 U の内部において、すべての対象の集合と射の集合の圏の圏になる。 すると通常の集合論の独立変数が、すべての圏の圏に適用される。さらに誤って固有類に対して言及する心配もなくなる。なぜならグロタンディーク宇宙は非常に広大であり、これはありとあらゆる数学的構造を充足させるからだ。
グロタンディーク宇宙において作業している場合、数学者はしばしば宇宙の公理を仮定する。"任意の集合 x に対し、x ∈U となるような宇宙 U が存在する。" この公理の重要な点は、任意の集合がいくつかの U に対して U-small が検討できることである。 つまり一般的なグロタンディーク宇宙に内部で、任意の独立変数が適用されるということである。この公理は強到達不能基数の存在と密接に関係している。
(>>385より) (参考:本スレより転載) Inter-universal geometry と ABC予想 52 https://rio2016.5ch.net/test/read.cgi/math/1588702281/606 (抜粋) 606 名前:132人目の素数さん[] 投稿日:2020/05/09(土) 17:57:38.91 ID:/BYRDNlz >案の定ZFCGがZFCの保存的拡大だのZFCの9個の公理だの間違えまくってる と書いてガゼで主張しただけだよ。 IUT-4で以下を修正しており、指摘で修正が必要であったのは、この箇所だよ。 concerning the “conservative extensionality” of ZFCG relative to ZFC, i.e., roughly speaking, that“any proposition that may be formulated in a ZFC-model and, moreover, holds in a ZFCG-model infact holds in the original ZFC-model” (引用終り)
それから ”On the basis of Axioms 1,3,4,5, define ⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) by” は、Axiom7〜9以降の式を簡便に記載するために用いる 記号⊆,Φ,S,∩,SING(x)の定義について語ってるだけ (Axiom6とは関係ないw)
The axiom schema of replacement is not necessary for the proofs of most theorems of ordinary mathematics. Indeed, Zermelo set theory (Z) already can interpret second-order arithmetic and much of type theory in finite types, which in turn are sufficient to formalize the bulk of mathematics. Although the axiom schema of replacement is a standard axiom in set theory today, it is often omitted from systems of type theory and foundation systems in topos theory.
(>>371より ww(^^; ) (参考) https://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf The Foundations of Mathematics Kenneth Kunen PDF 2007/10/29 - c 2005,2006,2007 Kenneth Kunen. Kenneth Kunen (抜粋)
P10 Axiom 6. Replacement Scheme. For each formula, φ, without B free, ∀x ∈ A∃!y φ(x, y) → ∃B ∀x ∈ A∃y ∈ B φ(x, y)
P11 Axiom 6. Replacement Scheme. For each formula, ψ, without B free, ∀x ∈ A∃!y ψ(x, y) → ∃B ∀x ∈ A∃y ∈ B ψ(x, y) The rest of the axioms are a little easier to state using some defined notions. On the basis of Axioms 1,3,4,5, define ⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) by: x ⊆ y ⇔ ∀z(z ∈ x → z ∈ y) x = Φ ⇔ ∀z(z not∈ x) y = S(x) ⇔ ∀z(z ∈ y ←→ z ∈ x ∨ z = x) w = x ∩ y ⇔ ∀z(z ∈ w ←→ z ∈ x ∧ z ∈ y) SING(x) ⇔ ∃y ∈ x ∀z ∈ x(z = y) (引用終り)
ここ誤読しているんじゃね? ”Although the axiom schema of replacement is a standard axiom in set theory today, it is often omitted from systems of type theory and foundation systems in topos theory.” ww(^^;
https://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf The Foundations of Mathematics Kenneth Kunen PDF 2007/10/29 - c 2005,2006,2007 Kenneth Kunen. Kenneth Kunen (抜粋) P10 Axiom 6. Replacement Scheme. For each formula, φ, without B free, ∀x ∈ A∃!y φ(x, y) → ∃B ∀x ∈ A∃y ∈ B φ(x, y) P11 Axiom 6. Replacement Scheme. For each formula, ψ, without B free, ∀x ∈ A∃!y ψ(x, y) → ∃B ∀x ∈ A∃y ∈ B ψ(x, y) The rest of the axioms are a little easier to state using some defined notions. On the basis of Axioms 1,3,4,5, define ⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) by: x ⊆ y ⇔ ∀z(z ∈ x → z ∈ y) x = Φ ⇔ ∀z(z not∈ x) y = S(x) ⇔ ∀z(z ∈ y ←→ z ∈ x ∨ z = x) w = x ∩ y ⇔ ∀z(z ∈ w ←→ z ∈ x ∧ z ∈ y) SING(x) ⇔ ∃y ∈ x ∀z ∈ x(z = y) (引用終り)
<補足> 1.確かに、これは ”define ⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) by:” で、これらを定義しているのだが 2.例えば、”x ⊆ y ⇔ ∀z(z ∈ x → z ∈ y)”で、⇔の右辺は On the basis of Axioms 1,3,4,5& Axiom 6. Replacement Scheme のみを組合わせて 左辺の ”x ⊆ y”が定義できます と読むべきもの 3.つまり、公理主義だか
https://en.wikipedia.org/wiki/Singleton_(mathematics) Singleton (mathematics) (抜粋) In mathematics, a singleton, also known as a unit set,[1] is a set with exactly one element. For example, the set {null?} is a singleton containing the element null. The term is also used for a 1-tuple (a sequence with one member).
https://ja.wikipedia.org/wiki/%E5%8D%98%E9%9B%86%E5%90%88 単集合 (抜粋) 数学における単集合(たんしゅうごう、英: singleton; 単元集合、単項集合、一元集合)あるいは単位集合(unit set[1])は、唯一の元からなる集合である。一つ組 (1-tuple) や単項列 (a sequence with one element) と言うこともできる。 例えば、{0} という集合は単集合である。
454 名前:の集合論的構成において、自然数の 1 とは単集合 {0} のことと定義される。 公理的集合論において、対の公理からの帰結として単元集合の存在が導かれる。即ち、任意の集合 A に対して、A と A に対して対の公理を適用すれば {A, A} なる集合の存在が保証されるが、これは A のみを元に持ちそれ以外の元は持たないから、単元集合 {A} に他ならない。 ここで A は任意の集合でよい、といっても集合がそもそもまったく存在しない場合には意味がないが、空集合の公理があれば少なくとも空集合 ? は集合になるから、A = ? ととって先の議論は正当化できる。 任意の集合 A と単集合 S に対し、A から S への写像はちょうど一つだけ存在する(それは A の各元を S の唯一の元へ写すものである)。従って任意の単元集合は集合の圏にける終対象である。 (引用終り) 以上 []
>>417 必死の論点ずらしご苦労さん (>>410再録) https://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf The Foundations of Mathematics Kenneth Kunen PDF 2007/10/29 Kenneth Kunen. Kenneth Kunen (抜粋) P10 Axiom 6. Replacement Scheme. For each formula, φ, without B free, ∀x ∈ A∃!y φ(x, y) → ∃B ∀x ∈ A∃y ∈ B φ(x, y) P11 Axiom 6. Replacement Scheme. For each formula, ψ, without B free, ∀x ∈ A∃!y ψ(x, y) → ∃B ∀x ∈ A∃y ∈ B ψ(x, y) The rest of the axioms are a little easier to state using some defined notions. On the basis of Axioms 1,3,4,5, define ⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) by: x ⊆ y ⇔ ∀z(z ∈ x → z ∈ y) x = Φ ⇔ ∀z(z not∈ x) y = S(x) ⇔ ∀z(z ∈ y ←→ z ∈ x ∨ z = x) w = x ∩ y ⇔ ∀z(z ∈ w ←→ z ∈ x ∧ z ∈ y) SING(x) ⇔ ∃y ∈ x ∀z ∈ x(z = y) (引用終り)
<補足> 1.確かに、これは ”define ⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) by:” で、これらを定義しているのだが 2.例えば、”x ⊆ y ⇔ ∀z(z ∈ x → z ∈ y)”で、⇔の右辺は On the basis of Axioms 1,3,4,5& Axiom 6. Replacement Scheme のみを組合わせて 左辺の ”x ⊆ y”が定義できます と読むべきもの 3.つまり、公理主義だから、公理で決められていないものを、天下りで 論理式 ψ で与えるわけにはいかないのです 迂遠でも、一歩一歩、ひとつづつ 公理を組合わせて ”x ⊆ y”などを えっちら おっちら 数学を展開するのに必要な定義を全て(のみならず全ての定理や命題)を 公理から 構築すべき or 構築できる それぞ、公理主義です
おサルは The rest of the axioms are a little easier to state using some defined notions. On the basis of Axioms 1,3,4,5, define ⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) by: x ⊆ y ⇔ ∀z(z ∈ x → z ∈ y) x = Φ ⇔ ∀z(z not∈ x) y = S(x) ⇔ ∀z(z ∈ y ←→ z ∈ x ∨ z = x) w = x ∩ y ⇔ ∀z(z ∈ w ←→ z ∈ x ∧ z ∈ y) SING(x) ⇔ ∃y ∈ x ∀z ∈ x(z = y)
これを x ⊆ y def⇒ ∀z(z ∈ x → z ∈ y) x = Φ def⇒ ∀z(z not∈ x) y = S(x) def⇒ ∀z(z ∈ y ←→ z ∈ x ∨ z = x) w = x ∩ y def⇒ ∀z(z ∈ w ←→ z ∈ x ∧ z ∈ y) SING(x) def⇒ ∃y ∈ x ∀z ∈ x(z = y)
と いうように 読んだらしいな つまり、左辺のx ⊆ y などを、「∀z(z ∈ x → z ∈ y)」と定義する その「∀z(z ∈ x → z ∈ y)」は、天下りに与えられるものだと
確かに、普通の数学本ならそうかも しかし、Kenneth Kunenの”The Foundations of Mathematics”は公理的集合論を説く教科書であり ZFC公理系から、いかに集合論を構築するのか? という視点で読むべきもの
SING(x) (x is a singleton) も、与えられた公理から導くか、さもなければ 最初から公理として与えるか? 二択しかありません! (多分w、ZFCでは ”singleton”の存在は 公理から導かれると思います。Kenneth Kunenの”The Foundations of Mathematics”には、そう書いてあるようですよw)
>>418 追加 (>>410再録) https://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf The Foundations of Mathematics Kenneth Kunen PDF 2007/10/29 Kenneth Kunen. Kenneth Kunen (抜粋) P10 On the basis of Axioms 1,3,4,5, define ⊆ (subset), Φ (or 0; empty set), S (ordinal successor function ), ∩ (intersection), and SING(x) (x is a singleton) by: x = Φ ⇔ ∀z(z not∈ x) >>309より (参考) https://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf The Foundations of Mathematics Kenneth Kunen PDF 2007/10/29 P10 I.2 The Axioms Axiom 0. Set Existence. ∃x(x = x) Axiom 1. Extensionality. ∀z(z ∈ x ←→ z ∈ y) → x = y Axiom 2. Foundation. ∃y(y ∈ x) → ∃y(y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y)) Axiom 3. Comprehension Scheme. For each formula, φ, without y free, ∃y∀x(x ∈ y ←→ x ∈ z ∧ φ(x)) Axiom 4. Pairing. ∃z(x ∈ z ∧ y ∈ z) Axiom 5. Union. ∃A∀Y ∀x(x ∈ Y ∧ Y ∈ F → x ∈ A) Axiom 6. Replacement Scheme. For each formula, φ, without B free, ∀x ∈ A∃!y φ(x, y) → ∃B ∀x ∈ A∃y ∈ B φ(x, y) Axiom 7. Infinity. ∃x({} ∈ x ∧ ∀y ∈ x(S(y) ∈ x))注:{}は空集合 Axiom 8. Power Set. ∃y∀z(z ⊆ x → z ∈ y) Axiom 9. Choice. {} not∈ F ∧ ∀x ∈ F ∀y ∈ F(x ≠ y → x ∩ y = {}) → ∃C ∀x ∈ F(SING(C ∩ x)) 注:{}は空集合 ZFC = Axioms 1-9. ZF = Axioms 1-8.
ZFC = Axioms 1–9. ZF = Axioms 1–8. ZC and Z are ZFC and ZF, respectively, with Axiom 6 (Replacement) deleted. Z −, ZF −, ZC −, ZFC − are Z , ZF, ZC , ZFC, respectively, with Axiom 2 (Foundation) deleted
Most of elementary mathematics takes place within ZC − (approximately, Zermelo’s theory). The Replacement Axiom allows you to build sets of size ℵω and bigger. It also lets you represent well-orderings by von Neumann ordinals, which is notationally useful, although not strictly necessary.
https://en.wikipedia.org/wiki/Well-founded_relation Well-founded relation (抜粋) Induction and recursion When the well-founded relation is set membership on the universal class, the technique is known as ∈-induction.
https://en.wikipedia.org/wiki/Epsilon-induction Epsilon-induction (抜粋) In mathematics, {\displaystyle \in }\in -induction (epsilon-induction or set-induction) is a variant of transfinite induction. Considered as an alternative set theory axiom schema, it is called the Axiom (schema) of (set) induction. It can be used in set theory to prove that all sets satisfy a given property P(x). This is a special case of well-founded induction.
>>465 コメントありがとう 9個の話は、下記 ”[i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].” で、”the nine axioms”に言い掛かりつけて、「望月先生が、基礎論・集合論を分かってない」というに
(参考) 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 P67 Section 3: Inter-universal Formalism: the Language of Species In the following discussion, we shall work with various models - consisting of “sets” and a relation “∈” - of the standard ZFC axioms of axiomatic set theory [i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].
https://en.wikipedia.org/wiki/Stack_(mathematics) Stack (mathematics) (抜粋) In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations of situations where isomorphic, compatible geometrical objects (such as vector bundles on topological spaces) can be "glued together" within a restriction of the topological basis. In a more general set-up the restrictions are replaced with pullbacks; fibred categories then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topology. Thus a stack is formally given as a fibred category over another base category, where the base has a Grothendieck topology and where the fibred category satisfies a few axioms that ensure existence and uniqueness of certain gluings with respect to the Grothendieck topology. Contents 1 Overview 2 Motivation and history 3 Definitions 4 Examples 5 Quasi-coherent sheaves on algebraic stacks 6 Other types of stack
Stack関連 Gerbe ”They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group.” (参考) https://en.wikipedia.org/wiki/Gerbe Gerbe (抜粋) In mathematics, a gerbe (/d???rb/; French: [???b]) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (Giraud 1971) following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them.
"Gerbe" is a French (and archaic English) word that literally means wheat sheaf.
History Gerbes first appeared in the context of algebraic geometry. They were subsequently developed in a more traditional geometric framework by Brylinski (Brylinski 1993). One can think of gerbes as being a natural step in a hierarchy of mathematical objects providing geometric realizations of integral cohomology classes.
A more specialised notion of gerbe was introduced by Murray and called bundle gerbes. Essentially they are a smooth version of abelian gerbes belonging more to the hierarchy starting with principal bundles than sheaves. Bundle gerbes have been used in gauge theory and also string theory. Current work by others is developing a theory of non-abelian bundle gerbes.
https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/214800/1/2004-03.pdf Title Deformation theory of algebraic stacks and its applications Author(s) 青木, 昌雄 Citation 代数幾何学シンポジューム記録 (2004), 2004: 20-29 Issue Date 2004
https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/232894/1/B51-07.pdf Title Quasi-coherent sheaves on algebraic moduli stacks of log structures (Algebraic Number Theory and Related Topics (2012) Author(s) Nagasaka, Tomohiro Citation 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu (2014), B51: 107-125 (引用終り) 以上
(参考) 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 P74 Remark 3.3.1. (i) One well-known consequence of the axiom of foundation of axiomatic set theory is the assertion that “∈-loops” a ∈ b ∈ c ∈ ... ∈ a can never occur in the set theory in which one works. On the other hand, there are many situations in mathematics in which one wishes to somehow “identify” mathematical objects that arise at higher levels of the ∈-structure of the set theory under consideration with mathematical objects that arise at lower levels of this ∈-structure.
In some sense, the notions of a “set” and of a “bijection of sets” allow one to achieve such “identifications”. That is to say, the mathematical objects at both higher and lower levels of the ∈-structure constitute examples of the same mathematical notion of a “set”, so that one may consider “bijections of sets” between those sets without violating the axiom of foundation. In some sense, the notion of a species may be thought of as a natural extension of this observation. That is to say, the notion of a “species” allows one to consider, for instance, speciesisomorphisms between species-objects that occur at different levels of the ∈-structure of the set theory under consideration - i.e., roughly speaking, to “simulate ∈-loops” - without violating the axiom of foundation. Moreover, typically the sorts of species-objects at diff
540 名前:erent levels of the ∈-structure that one wishes to somehow have “identified” with one another occur as the result of executing the mutations that arise in some sort of mutation-history ... → S → S_ → S_ _ → ... → S ... [where S = (S0, S1); S_ = (S0_, S1_); S_ _ = (S0_ _, S1_ _) are species] - e.g., the “output species-objects” of the “S” on the right that arise from applying various mutations to the “input species-objects” of the “S” on the left.
(ii) In the context of constructing “loops” in a mutation-history as in the final display of (i), we observe that the simpler the structure of the species involved, the easier it is to construct “loops”. It is for this reason that species such as the species determined by the notion of a category [cf. Example 3.2] are easier to work with, from the point of view of constructing “loops”, than more complicated species such as the species determined by the notion of a scheme. This is one of the principal motivations for the “geometry of categories” - of which “absolute anabelian geometry” is the special case that arises when the categories involved are Galois categories - i.e., for the theory of representing scheme-theoretic geometries via categories [cf., e.g., the Introductions of [MnLg], [SemiAnbd], [Cusp], [FrdI]]. At a more concrete level, the utility of working with categories to reconstruct objects that occurred at earlier stages of some sort of “series of constructions” [cf. the mutation-history of the final display of (i)!] may be seen in the “reconstruction of the underlying scheme” in various situations throughout [MnLg] by applying the natural equivalence of categories of the final display of [MnLg], Definition 1.1, (iv), from a certain category constructed from a log scheme, as well as in the theory of “slim exponentiation” discussed in the Appendix to [FrdI].
(iii) Again in the context of mutation-histories such as the one given in the final display of (i), although one may, on certain occasions, wish to apply various mutations that fundamentally alter the structure of the mathematical objects involved and hence give rise to “output species-objects” of the “S” on the right that are related in a highly nontrivial fashion to the “input species-objects” of the “S”on the left, it is also of interest to consider “portions” of the various mathematical objects that occur that are left unaltered by the various mutations that one applies. This is precisely the reason for the introduction of the notion of a core of a mutationhistory. One important consequence of the construction of various cores associated to a mutation-history is that often one may apply various cores associated to a mutation-history to describe, by means of non-coric observables, the portions of the various mathematical objects that occur which are altered by the various mutations that one applies in terms of the unaltered portions, i.e., cores. Indeed, this point of view plays a central role in the theory of the present series of papers - cf. the discussion of Remark 3.6.1, (ii), below. (引用終り) 以上
このIUT IV P74 Remark 3.3.1. をコピーしたのは “∈-loops”とか、”∈-structure of the set theory”とか ”In the context of constructing “loops” in a mutation-history”、”“output species-objects” of the “S” on the right that” とか
1.長い 読めない証明のギネスは、有限単純群の分類定理です。”which is probably around 10000 to 20000 pages.”と言われる 2.ここのほんの一部ですが、”Quasithin groups The classification of the simple quasithin groups by Aschbacher and Smith was 1221 pages long, one of the longest single papers ever written.”です Quasithinというのは、和訳では「準薄」とか書かれることが多いようですが、この部分だけで 1221 pagesだとか。IUT I〜IV 計600ページの2倍です 3.で、この有限単純群の分類定理 を全部読んだ人は、おそらく居ない!w (多分 「準薄」の1221 pagesだって、読む人は ほんの小数でしょうね(この1221 pagesは、おそらく特殊分野で 論文としては孤立していて、他にはあまり使えないかも)) 4.しかし、「有限単純群の分類定理」は、納得性があるのです。ここでは書きませんが。解説書も、何冊か出ています。(私は、岩波の鈴木通夫先生の上下2冊を読みましたけど) 5.いずれ、IUTもそうなると思います。(準備論文を含めると、数千ページなのでしょう。納得性のある解説が、求められます(^^; )
(参考) https://en.wikipedia.org/wiki/List_of_long_mathematical_proofs List of long mathematical proofs (抜粋) As of 2011, the longest mathematical proof, measured by number of published journal pages, is the classification of finite simple groups with well over 10000 pages. There are several proofs that would be far longer than this if the details of the computer calculations they depend on were published in full.
・2004 Quasithin groups The classification of the simple quasithin groups by Aschbacher and Smith was 1221 pages long, one of the longest single papers ever written. ・2004 Classification of finite simple groups. The proof of this is spread out over hundreds of journal articles which makes it hard to estimate its total length, which is probably around 10000 to 20000 pages.
(参考) https://en.wikipedia.org/wiki/Quasithin_group Quasithin group (抜粋) Classification The classification of quasithin groups is a crucial part of the classification of finite simple groups. The quasithin groups were classified in a 1221-page paper by Michael Aschbacher and Stephen D. Smith (2004, 2004b). An earlier announcement by Geoffrey Mason (1980) of the classification, on the basis of which the classification of finite simple groups was announced as finished in 1983, was premature as the unpublished manuscript (Mason 1981) of his work was incomplete and contained serious gaps.
References ・Mason, Geoffrey (1980), "Quasithin groups", in Collins, Michael J. (ed.), Finite simple groups. II, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], pp. 181?197, ISBN 978-0-12-181480-9, MR 0606048 ・Mason, Geoffrey (1981), The classification of finite quasithin groups, U. California Santa Cruz, p. 800 (unpublished typescript)
有名どころでは、”1799 The Abel?Ruffini theorem was nearly proved by Paolo Ruffini, but his proof, spanning 500 pages, was mostly ignored and later, in 1824, Niels Henrik Abel published a proof that required just six pages.” これは、5次の代数方程式の 代数的解法が存在しないことの定理ですが、Paolo Ruffiniが500ページほど書いて証明したそうですが(ギャップがあったとか)、アーベルが” just six pages”にしたとか
(参考) https://en.wikipedia.org/wiki/List_of_long_mathematical_proofs List of long mathematical proofs (抜粋) Long proofs The length of unusually long proofs has increased with time. As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof. ・1799 The Abel?Ruffini theorem was nearly proved by Paolo Ruffini, but his proof, spanning 500 pages, was mostly ignored and later, in 1824, Niels Henrik Abel published a proof that required just six pages. ・1964 Resolution of singularities Hironaka's original proof was 216 pages long; it has since been simplified considerably down to about 10 or 20 pages. ・1966 Abyhankar's proof of resolution of singularities for 3-folds in characteristic greater than 6 covered about 500 pages in several papers. In 2009, Cutkosky simplified this to about 40 pages. ・1966 Discrete series representations of Lie groups. Harish-Chandra's construction of these involved a long series of papers totaling around 500 pages. His later work on the Plancherel theorem for semisimple groups added another 150 pages to these. ・1960?1970 Fondements de la Geometrie Algebrique, Elements de geometrie algebrique and Seminaire de geometrie algebrique. Grothendieck's work on the
570 名前:foundations of algebraic geometry covers many thousands of pages. Although this is not a proof of a single theorem, there are several theorems in it whose proofs depend on hundreds of earlier pages. []
>>528 >・1964 Resolution of singularities Hironaka's original proof was 216 pages long; it has since been simplified considerably down to about 10 or 20 pages.
これは、有名なHironaka先生の特異点解消定理ですが、216 pagesを ”about 10 or 20 pages”に出来たとか 望月先生は、否定するかもしれませんが、短くできる可能性は否定できないでしょう
https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ Persiflage=Frank Calegari氏シカゴ大 The ABC conjecture has (still) not been proved Posted on December 17, 2017 (抜粋) Terence Tao says: December 18, 2017 at 2:46 pm Thanks for this. I do not have the expertise to have an informed first-hand opinion on Mochizuki’s work, but on comparing this story with the work of Perelman and Yitang Zhang you mentioned that I am much more familiar with, one striking difference to me has been the presence of short “proof of concept” statements in the latter but not in the former, by which I mean ways in which the methods in the papers in question can be used relatively quickly to obtain new non-trivial results of interest (or even a new proof of an existing non-trivial result) in an existing field.
From what I have read and heard, I gather that currently, the shortest “proof of concept” of a non-trivial result in an existing (i.e. non-IUTT) field in Mochizuki’s work is the 300+ page argument needed to establish the abc conjecture. It seems to me that having a shorter proof of concept (e.g. <100 pages) would help dispel scepticism about the argument. It seems bizarre to me that there would be an entire self-contained theory whose only external application is to prove the abc conjecture after 300+ pages of set up, with no smaller fragment of this setup having any non-trivial external consequence whatsoever.
1.望月論文には、Perelmanなどの論文にある 概念の説明が無い(不足している?) 2.”short “proof of concept” statements”があれば、他の分野にも応用できる 3.300ページ以上使って、ようやく 「“proof of concept” of a non-trivial result in an existing (i.e. non-IUTT) field」 になる 4.もっと短くできるんじゃね?w 「It seems to me that having a shorter proof of concept (e.g. <100 pages) would help dispel scepticism about the argument.」 5.300ページ以上の設定の後に、abcの仮定を証明することだけが唯一の外部応用である自己完結型の理論が存在することは、私には奇妙なことのように思えます。 「It seems bizarre to me that there would be an entire self-contained theory whose only external application is to prove the abc conjecture after 300+ pages of set up, with no smaller fragment of this setup having any non-trivial external consequence whatsoever.」
590 名前:シスレからだが 転載しておく(^^ https://rio2016.5ch.net/test/read.cgi/math/1582599485/114-115 https://ja.wikipedia.org/wiki/%E3%83%AC%E3%82%A4%E3%83%A2%E3%83%B3%E3%83%89%E3%83%BB%E3%82%B9%E3%83%9E%E3%83%AA%E3%83%A4%E3%83%B3 レイモンド・メリル・スマリヤン(Smullyan、1919年5月25日 - 2017年2月6日)は合衆国の数学者 ニューヨーク市のFar Rockawayに生れる。最初は奇術師をしていた。1955年にシカゴ大学から学士を得る。1959年にプリンストン大学から博士号を得る。アロンゾ・チャーチのもとで学んだ数多くの傑出した論理学者の一人 経歴 スマリヤンは博士課程にいるときの1957年に“Journal of Symbolic Logic”に論文を発表し、ゲーデルの不完全性定理が1931年にゲーデルが発表した論文よりも初等的な形で形式系を考察できることを示した ゲーデルの不完全性定理に関する現代的な解釈はこの論文から始まっている。その後、スマリヤンはゲーデルの不完全性定理における魅力的な部分がタルスキの定理から必然的に導かれることを示した タルスキの定理は不完全性定理よりも容易に証明できて、哲学的に不完全性定理と同じような不安を与えるものである 数理論理学において古典的な限界を与える定理に関してスマリヤンが終生寄与した成果は以下の文献で読むことができる: ・Smullyan, R M (2001) "Godel's Incompleteness Theorems" in Goble, Lou, ed.,
(PDFが落とせる) https://www.researchgate.net/publication/230961342_Godel_incompleteness_theorems_and_the_limits_of_their_applicability_I Godel incompleteness theorems and the limits of their applicability. I Article in Russian Mathematical Surveys 65(5):857 ・ January 2011 Lev Dmitrievich Beklemishev Abstract This is a survey of results related to the Godel incompleteness theorems and the limits of their applicability. The first part of the paper discusses Godel's own formulations along with modern strengthenings of the first incompleteness theorem. Various forms and proofs of this theorem are compared. Incompleteness results related to algorithmic problems and mathematically natural examples of unprovable statements are discussed. Bibliography: []
(参考) https://ja.wikipedia.org/wiki/%E3%82%AF%E3%83%AC%E3%83%AC%E8%AA%8C クレレ誌 歴史 この学術雑誌はオーガスト・レオポルト・クレレにより、1826年に創刊され、1855年に彼が亡くなるまで、クレレによって編纂された。 クレレ誌はアカデミーの紀要ではない最初の主要な数学学術誌の一つである(Neuenschwander 1994, p. 1533)。ニールス・アーベル、ゲオルク・カントール、ゴットホルト・アイゼンシュタインらの研究を含む著名な論文を掲載してきた。
https://ja.wikipedia.org/wiki/%E7%8E%8B%E7%AB%8B%E5%AD%A6%E4%BC%9A 王立学会 王立学会(おうりつがっかい)は、1660年にロンドンで作られた民間の科学に関する団体であるthe Royal Society of Londonのことである[1]。
出版物 フィロソフィカル・トランザクションズ1665年版の表紙 学会の機関誌として、「フィロソフィカル・トランザクションズ 」(The Philosophical Transactions of the Royal Society)がある。 発会時からメンバーだったヘンリー・オルデンバーグ(1619-1677)は初代事務総長で、科学者間の実験哲学や数理哲学に関する情報ネットワークの構築に尽力した。 オルデンバーグは情報発信のために個人の費用でこの雑誌を1665年に創刊した。数年後に学会の刊行物となった[21]。
「GREEN JUKEBOX」編は、これまでロックバンド「SEKAI NO OWARI」(セカオワ)のFukaseさん、「RADWIMPS(ラッドウィンプス)」の野田洋次郎さん、「back number(バックナンバー)」の清水依与吏(いより)さんらが出演し、自身の人気曲のアコースティックバージョンを披露してきた。
型無しラムダ計算 20年ほど時代を下り,1930年代に話題を移そう.若き日の Alonzo Church は,自由変数を用いない*19形式論理学の記法あるいは計算体系として,ラムダ計算を提案する. 初出は1932年の A Set of Postulates for the Foundation of Logic であるようで,表題からも分かる通り,この頃の Church はラムダ計算を論理学の基礎として据えようと考えていたらしく,項として種々の論理定項を含んでいる. しかし,このオリジナルの体系は証明力が強すぎたため,後に Stephen Kleene と John Barkley Rosser らにより矛盾を導くことが証明された*20.
単純型付ラムダ計算 1940年に発表された A Formulation of the Simple Theory of Types という論文が,型付きラムダ計算の初出とされている. ラムダ計算と階型理論を統合することで,どのような嬉しい性質が生じるのかはこの論文には記されていないが,ともかく Church は 従来のラムダ計算に加えて,型という概念を導入する.
終わりに これまで見てきたように,制限のない自己言及はしばしば矛盾を引き起こす. これを防ぐために Russell は型の概念を発明し,論理学における式がどのような文脈で出現できるかに制限を加え,Church はこのアイディアを拝借し,ラムダ計算における式がどのような文脈で出現できるかに制限を加えた.
Russell の型理論も Church のラムダ計算も,論理学,ひいては数学の基礎付けを与えるという願いを成就することはできなかった. しかし,これらが組み合わさった際に,プログラムと証明との間に対応関係が浮かび上がり,再び論理学の世界へと通じる道が開くことは Curry-Howard 同型対応としてよく知られている. 結局のところ,両者の思想ともに,人間の思考の根源的な営みである論理や推論と言った概念にどこか根ざしているように思われて仕方ないのである. (引用終り) 以上
Restriction of syntax: つまりラッセル集合の定義文は「文法違反」だ、というもの Restriction of logic: つまりパラドックスは古典論理のせいだ、 だから古典論理を制限/変更しようというもの Restriction of basic principles: つまり包括原理が問題だ、というもの
同意(^^ 1)IUTの「同義反復的解決」(>>603)が、ラッセルのパラドックス同様の自己言及(二階述語)と見て 2)「基礎付けの公理は、ラッセルのパラドックスを防ぐために導入された」という俗説に 引っ掛かって 3)基礎付けの公理に絡んで、“∈-structure”とか あるいは ”the notion of a species allows one to “simulate ∈-loops” without violating the axiom of foundation of axiomatic set theory - cf. the discussion of Remark 3.3.1, (i).” などと 言い訳を言っている気がする 俗説に 引っ掛かってw(^^;
677 名前:NTER-UNIVERSAL TEICHMULLER THEORY IV: ¨ LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki April 2020 P1 Abstract Finally, we examine - albeit from an extremely naive/non-expert point of view! - the foundational/settheoretic issues surrounding the vertical and horizontal arrows of the log-theta-lattice by introducing and studying the basic properties of the notion of a “species”, which may be thought of as a sort of formalization, via set-theoretic formulas, of the intuitive notion of a “type of mathematical object”. These foundational issues are closely related to the central role played in the present series of papers by various results from absolute anabelian geometry, as well as to the idea of gluing together distinct models of conventional scheme theory, i.e., in a fashion that lies outside the framework of conventional scheme theory. 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”.
P3 Finally, in §3, we examine - albeit from an extremely naive/non-expert point of view! - certain foundational issues underlying the theory of the present series of papers. Typically in mathematical discussions [i.e., by mathematicians who are not equipped with a detailed knowledge of the theory of foundations!] - such as, for instance, the theory developed in the present series of papers! - one defines various “types of mathematical objects” [i.e., such as groups, topological spaces, or schemes], together with a notion of “morphisms” between two particular examples of a specific type of mathematical object [i.e., morphisms between groups, between topological spaces, or between schemes].
P6 Indeed, from the point of view of the “∈-structure” of axiomatic set theory, there is no way to treat sets constructed at distinct levels of this ∈-structure as being on a par with one another. On the other hand, if one focuses not on the level of the ∈-structure to which a set belongs, but rather on species, then the notion of a species allows one to relate - i.e., to treat on a par with one another - objects belonging to the species that arise from sets constructed at distinct levels of the ∈-structure. That is to say, the notion of a species allows one to “simulate ∈-loops” without violating the axiom of foundation of axiomatic set theory - cf. the discussion of Remark 3.3.1, (i).
P8 Acknowledgements: I would like to thank Kentaro Sato for useful comments concerning the set-theoretic and foundational aspects of the present paper,
P67 Section 3: Inter-universal Formalism: the Language of Species In the present §3, we develop - albeit from an extremely naive/non-expert point of view, relative to the theory of foundations! - the language of species. Roughly speaking, a “species” is a “type of mathematical object”, such as a “group”, a “ring”, a “scheme”, etc.
In some sense, this language may be thought of as an explicit description of certain tasks typically executed at an implicit, intuitive level by mathematicians [i.e., mathematicians who are not equipped with a detailed knowledge of the theory of foundations!] via a sort of “mental arithmetic” in the course of interpreting various mathematical arguments. In the context of the theory developed in the present series of papers, however, it is useful to describe these intuitive operations explicitly
P68 - where n ranges over the natural numbers. On the other hand, by the axiom of foundation, there do not exist infinite descending chains of universes V0 ∋ V1 ∋ V2 ∋ V3 ∋ ... ∋ Vn ∋ ... - where n ranges over the natural numbers.
P74 Remark 3.3.1. (i) One well-known consequence of the axiom of foundation of axiomatic set theory is the assertion that “∈-loops” a ∈ b ∈ c ∈ ... ∈ a can never occur in the set theory in which one works. On the other hand, there are many situations in mathematics in which one wishes to somehow “identify” mathematical objects that arise at higher levels of the ∈-structure of the set theory under consideration with mathematical objects that arise at lower levels of this ∈-structure. In some sense, the notions of a “set” and of a “bijection of sets” allow one to achieve such “identifications”. That is to say, the mathematical objects at both higher and lower levels of the ∈-structure constitute examples of the same mathematical notion of a “set”, so that one may consider “bijections of sets” between those sets without violating the axiom of foundation. In some sense, the notion of a species may be thought of as a natural extension of this observation. That is to say, the notion of a “species” allows one to consider, for instance, speciesisomorphisms between species-objects that occur at different levels of the ∈-structure of the set theory under consideration - i.e., roughly speaking, to “simulate ∈-loops” - without violating the axiom of foundation. (引用終り) 以上
>>624 補足 > ”the notion of a species allows one to “simulate ∈-loops” without violating the axiom of foundation of axiomatic set theory - cf. the discussion of Remark 3.3.1, (i).”
1.”violating the axiom of foundation ”は、当然ながら、公理系の中では許されない 2.では、”the notion of a species allows one to “simulate ∈-loops””とは、具体的に何なのか? IUTでどういう役割を果たすのか? きっと重要なのでしょうね、望月先生は一生懸命書いてあるから 3.なんとなく、”the notion of a species allows one to “simulate ∈-loops””を成り立たせるための仕掛けが、IUT IIIまでにしてあって そこ、ショルツ先生に理解されなかったのでは? きっと、奇妙に見えることをしているのでは? で、ショルツ先生から「奇妙にしか見えない!」と ずばり指摘されて 「おまえは 分かっていない!!」と つい叫んでしまった 望月先生だった
https://ja.wikipedia.org/wiki/%E7%92%B0%E3%81%AE%E5%9C%8F 環の圏 (抜粋) 数学の特に圏論における(単位的・結合)環の圏(かんのけん、英: category of rings)Ring は、すべての(単位元持つ)環を対象とし、すべての(単位元を保つ)環準同型を射とする圏である。他の多くの例と同じく、環の圏は大きい(すなわち、すべての環の成す類は集合でない真の類である)。
具体圏として 環の圏 Ring は具体圏(英語版)、すなわちその対象は集合に追加の構造(いまの場合、加法と乗法)を入れたものであり、その射はそれら構造を保つ写像である。環の圏から集合の圏への自然な忘却函手(英語版) U: Ring → Set が、各環をその台となる集合へ写すことによって(つまり、加法と乗法という演算を「忘れる」ことによって)与えられる。 この忘却函手の左随伴 F: Set → Ring は各集合 X に X の生成する自由環を対応させる自由函手である。
環の圏を、アーベル群の圏 Ab 上の、あるいはモノイドの圏(英語版)[要リンク修正] Mon 上の具体圏と見ることもできる。具体的に、乗法あるいは加法をそれぞれ忘れることによって、二つの忘却函手 A: Ring → Ab および M: Ring → Mon が得られる(つまりA は環の加法群を取り出す函手、M は環の吸収元付き乗法モノイドを取り出す函手である)。 この二つはいずれも左随伴を持つ。A の左随伴は、任意のアーベル群 X に対し(それを Z-加群と見て)テンソル環 T(X) を割り当てる函手である。また M の左随伴は、任意のモノイド G に整係数モノイド環 Z[G] が対応する。
射について 数学においてよく知られた多くの圏と異なり、環の圏 Ring の任意の二対象の間には必ずしも射が存在するわけではない。これは(単位的)環準同型が単位元を保つという事実の反映である。例えば、零環 0 = {0} から任意の非零環への射は存在しない。環 R から S への射が存在するためには、S の標数が R の標数を割り切ることが必要条件である。
射集合が空となることがあってさえ、それでも始対象が存在するから、環の圏 Ring は連結(英語版)である。
部分圏について 環の圏 Ring はいくつも重要な部分圏を持っている。例えば、可換環、整域、主イデアル環、体それぞれの全体の成す充満部分圏などが挙げられる。
体の圏 体の圏 Field は、すべての可換体を対象とする CRing の充満部分圏である。体の圏はほかの代数圏のようにはよく振る舞わない。特に「自由体」(すなわち忘却函手 Field → Set の左随伴となるもの)は存在しない。したがって、Field は CRing の反映的部分圏ではない。
体の圏 Field は有限完備(英語版)でも有限余完備でもない。特に、Field は積も余積も持たない。
もう一つ体の圏 Field の著しい点は、任意の射が単型射となることである。
アーベル群の圏 Ab から擬環の圏 Rng への忠実充満函手が、各アーベル群を、それに自明な積を入れた零擬環に対応させることで与えられる。 (引用終り) 以上
動機 環はその環上の加群を通じて研究されることが一般的である。これは加群が環の表現と見做せるからである。すべての環 R は環の積による作用によって自然に R 加群の構造を持つので、加群論的な研究方法はより一般的で有益な情報をもたらす。このような訳で、環についての研究はその環上の加群の成す圏を研究することによってしばしば為される。
https://en.wikipedia.org/wiki/Morita_equivalence Morita equivalence (抜粋) Significance in K-theory If two rings are Morita equivalent, there is an induced equivalence of the respective categories of projective modules since the Morita equivalences will preserve exact sequences (and hence projective modules). Since the algebraic K-theory of a ring is defined (in Quillen's approach) in terms of the homotopy groups of (roughly) the classifying space of the nerve of the (small) category of finitely generated projective modules over the ring, Morita equivalent rings must have isomorphic K-groups. 以上
IUT IV(>>624) ”the notion of a species allows one to “simulate ∈-loops” without violating the axiom of foundation of axiomatic set theory - cf. the discussion of Remark 3.3.1, (i).” の関係というか、繋がりを探しているが
あんまり関係ないが、メモ貼る https://en.wikipedia.org/wiki/Algebraic_combinatorics Algebraic combinatorics (抜粋) Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
Contents 1 History 2 Scope 3 Important topics 3.1 Symmetric functions 3.2 Association schemes 3.3 Strongly regular graphs 3.4 Young tableaux 3.5 Matroids 3.6 Finite geometries
History The term "algebraic combinatorics" was introduced in the late 1970s.[1] Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux). This period is reflected in the area 05E, Algebraic combinatorics, of the AMS Mathematics Subject Classification, introduced in 1991.
Scope Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group and representation theory, lattice theory and commutative algebra are common.
下記の Lawvere Ph.D. thesis は、一読の価値あるな ”The authors comments are F. William Lawvere, 2004.”の部分
710 名前:だけでも、読んでおく価値がある!(^^ http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories F. William Lawvere Originally published as: Ph.D. thesis, Columbia University, 1963 and in Reports of the Midwest Category Seminar II, 1968, 41-61, The authors comments are F. William Lawvere, 2004. http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf
https://en.wikipedia.org/wiki/Universal_algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study. History Starting with William Lawvere's thesis in 1963, techniques from category theory have become important in universal algebra.[9] Footnotes 9 Lawvere, William F. (1964), Functorial Semantics of Algebraic Theories (PhD Thesis)
https://en.wikipedia.org/wiki/Category_theory Category theory Historical notes Main article: Timeline of category theory and related mathematics https://en.wikipedia.org/wiki/Timeline_of_category_theory_and_related_mathematics Category theory may be viewed as an extension of universal algebra, as the latter studies algebraic structures, and the former applies to any kind of mathematical structure and studies also the relationships between structures of different nature. []
それ 意味分かんないけど S・Sレポ8月版 って、2018年でしょ? そして、2020年4月3日に「査読終り。IUT成立。SSレポートはアウト」の記者会見 その後、Woitブログで、ショルツ先生 vs Dupuy先生のバトルが、下記
https://www.math.columbia.edu/~woit/wordpress/?p=11709 (woitブログ) Not Even Wrong Latest on abc Posted on April 3, 2020 by woit
より、以下抜粋
1) 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.)
2) Peter Scholze says: April 30, 2020 at 3:32 am Dear Taylor,
I certainly understood your point there ? you might also take the ring Z[√(-1)].
There is of course a big difference between the ring Z and the “theory” it defines, i.e. roughly the class of all subsets of all finite powers Z^n that are definable by polynomial equations. The latter is indeed a highly nontrivial category (where morphisms are given by definable graphs); it is of course not equivalent to a category with one object and two morphisms. If a category like this is in place in Mochizuki’s work, I’m happy to hear about it!
Reading the IUT papers, however, you are presented with some extremely difficult notion of a Hodge theater, together with a highly non-obvious notion of isomorphisms of such: Isomorphisms do not preserve nearly as much structure as you would expect them to, and this is by design as Mochizuki points out. So I find it very hard to “guess” what something like a surrounding “theory” might be. For all I can see, Hodge theaters fit neither into the framework of “structures” as used in the wikipedia entry https://en.wikipedia.org/wiki/Interpretation_(model_theory) you linked to, nor the topos-theoretic framework of Caramello. (Regarding the first one: A “structure” in the sense of model theory has first of all an underlying set. I find it hard to take a Hodge theater and produce some interesting set that is functorial in isomorphisms of Hodge theaters, the problem being the very lax notion of isomorphisms of Hodge theaters.)
However, these long discussions are all about interpretations. Regarding the mathematics proper: I stand by the claim made in our manuscript, and have indicated the proof above.
1.ショルツ先生も、望月IUT難しすぎで、分からないことがある 2.でも、自分が正しいと思う(数学的には証明になってないみたいw(^^; ) 3.Dupuy先生納得せず。”I’m happy to continue any further discussions by e-mail.”(あとは、メールで)と ショルツ先生
https://en.wikipedia.org/wiki/Arithmetic_geometry In the 1960s, Goro Shimura introduced Shimura varieties as generalizations of modular curves.[20] Since the 1979, Shimura varieties have played a crucial role in the Langlands program as a natural realm of examples for testing conjectures.[21]
(参考) https://www.mathunion.org/icm/icm-2022 ICM 2022 The International Congress of Mathematicians 2022 (ICM 2022) will be held in St Petersburg, Russia, 6 - 14 July 2022.
下記のArithmetic geometry Mid-to-late 20th century: developments in modularity, p-adic methods, and beyond のその先にIUTが位置するのだろうと 見ています(^^
(参考) https://en.wikipedia.org/wiki/Arithmetic_geometry Arithmetic geometry (抜粋) Overview The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity.[5] The structure of algebraic varieties defined over non-algebraically-closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite fields, etale cohomology provides topological invariants associated to algebraic varieties.[6] p-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields.[7]
History Mid-to-late 20th century: developments in modularity, p-adic methods, and beyond
In 1983, Gerd Faltings proved the Mordell conjecture, demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell?Weil theorem only demonstrates finite generation of the set of rational points as opposed to finiteness).[25][26]
In 2001, the proof of the local Langlands conjectures for GLn was based on the geometry of certain Shimura varieties.[27]
In the 2010s, Peter Scholze developed perfectoid spaces and new cohomology theories in arithmetic geometry over p-adic fields with application to Galois representations and certain cases of the weight-monodromy conjecture.[28][29]
>>708 (抜粋) 3) Peter Scholze says: May 1, 2020 at 4:42 pm Dear Taylor, I’m happy to continue any further discussions by e-mail. (引用終り)
Taylor Dupuyの勝利に終わったのか? それは、ともかく Taylor Dupuyのカベは、ショルツ先生には破れないだろうな(^^;
(参考) math jin:(IUTT情報サイト) https://twitter.com/math_jin (抜粋) math_jinさんがリツイート Taylor Dupuy 5月3日 返信先: @Rz6rX さん, @math_jin さん 他2人 The manuscript with Stix has problems. The worst being the diagram in section 2.2. That does not appear in Mochizuki’s manuscript. I think people should stop citing it.
math_jinさんがリツイート 太郎 @Rz6rX 5月3日 返信先: @DupuyTaylor さん, @math_jin さん 他2人 Excuse me, It may have been asked many times, but is Professor Scholze's point true? (deleted an unsolicited ad)
(下記みたいだな) www.kurims.kyoto-u.ac.jp/~motizuki/papers-japanese.html 望月 論文 遠アーベル幾何、圏の幾何 [25] On the Combinatorial Anabelian Geometry of Nodally Nondegenerate Outer Representations. PDF NEW !! (2011-12-22) Comments NEW !! (2019-07-20)
798 名前:(Revised November 11, 2010)で、微妙に違う(^^; http://www.kurims.kyoto-u.ac.jp/~yuichiro/papers.html 星裕一の論文 組み合わせ論的遠アーベル幾何学関連 https://projecteuclid.org/download/pdf_1/euclid.hmj/1323700038 Hiroshima Math. J. 41 (2011), 275?342 On the combinatorial anabelian geometry of nodally nondegenerate outer representations Yuichiro Hoshi and Shinichi Mochizuki (Received August 19, 2009) (Revised November 11, 2010) Abstract. Let S be a nonempty set of prime numbers. In the present paper, we continue the study, initiated in a previous paper by the second author, of the combinatorial anabelian geometry of semi-graphs of anabelioids of pro-S PSC-type, i.e., roughly speaking, semi-graphs of anabelioids associated to pointed stable curves. Our first main result is a partial generalization of one of the main combinatorial anabelian results of this previous paper to the case of nodally nondegenerate outer representations, i.e., roughly speaking, a sort of abstract combinatorial group-theoretic generalization of the scheme-theoretic notion of a family of pointed stable curves over the spectrum of a discrete valuation ring. We then apply this result to obtain a generalization, to the case of proper hyperbolic curves, of a certain injectivity result, obtained in another paper by the second author, concerning outer automorphisms of the pro-S fundamental group of a configuration space associated to a hyperbolic curve, as the dimension of this configuration space is lowered from two to one. This injectivity allows one to generalize a certain well-known injectivity theorem of Matsumoto to the case of proper hyperbolic curves. (引用終り) 以上 []
https://arxiv.org/abs/1902.03767 [Submitted on 11 Feb 2019 (v1), last revised 24 Aug 2019 (this version, v3)] A topological approach to indices of geometric operators on manifolds with fibered boundaries Mayuko Yamashita In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries. We define K-groups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete Φ or edge metrics, can be regarded as the index pairing over these K-groups. We also prove various properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles. Comments: 69 pages; the final version Subjects: K-Theory and Homology (math.KT); Differential Geometry (math.DG)
(IUT IV P85 Bibliography より) [Drk] F. R. Drake, Set Theory: an Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, North-Holland (1974).
なお、”The program was founded by Harvey Friedman (1975, 1976) and brought forward by Steve Simpson. ” とあるように、学問というのは 大体が 創始者を踏み越えて、進んでいくものです(^^;
(参考) https://en.wikipedia.org/wiki/
910 名前:Reverse_mathematics Reverse mathematics
The program was founded by Harvey Friedman (1975, 1976) and brought forward by Steve Simpson. A standard reference for the subject is (Simpson 2009), while an introduction for non-specialists is (Stillwell 2018). An introduction to higher-order reverse mathematics, and also the founding paper, is (Kohlenbach (2005)).
Contents 1 General principles 1.1 Use of second-order arithmetic 1.2 Use of higher-order arithmetic 2 The big five subsystems of second-order arithmetic 2.1 The base system RCA0 2.2 Weak K?nig's lemma WKL0 2.3 Arithmetical comprehension ACA0 2.4 Arithmetical transfinite recursion ATR0 2.5 Π11 comprehension Π11-CA0 3 Additional systems 4 ω-models and β-models []
(参考) https://en.wikipedia.org/wiki/List_of_incomplete_proofs List of incomplete proofs
・Fundamental theorem of algebra (see History). Many incomplete or incorrect attempts were made at proving this theorem in the 18th century, including by d'Alembert (1746), Euler (1749), de Foncenex (1759), Lagrange (1772), Laplace (1795), Wood (1798), and Gauss (1799). The first rigorous proof was published by Argand in 1806.
・Dirichlet's principle. This was used by Riemann in 1851, but Weierstrass found a counterexample to one version of this principle in 1870, and Hilbert stated and proved a correct version in 1900.
・The proofs of the Kronecker?Weber theorem by Kronecker (1853) and Weber (1886) both had gaps. The first complete proof was given by Hilbert in 1896.
・In 1905 Lebesgu
913 名前:e tried to prove the (correct) result that a function implicitly defined by a Baire function is Baire, but his proof incorrectly assumed that the projection of a Borel set is Borel. Suslin pointed out the error and was inspired by it to define analytic sets as continuous images of Borel sets.
・ Class numbers of imaginary quadratic fields. In 1952 Heegner published a solution to this problem. His paper was not accepted as a complete proof as it contained a gap, and the first complete proofs were given in about 1967 by Baker and Stark. In 1969 Stark showed how to fill the gap in Heegner's paper. []
全くその通りです で、下記のList of long mathematical proofs を覗いてみて
30ほどリストがあるでしょ みんな、そうなのよ(長くて読めない)w(^^;
”The length of unusually long proofs has increased with time. As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof.” と書かれていますよね。論文が だんだん、長くなっていると
正直、下記のリストでも、もうすでに常人では読めない長さの論文があります ”around 10000 to 20000 pages.”は 別格としても、この”by Aschbacher and Smith was 1221 pages long”なんて、だれが読むんだ?ってw
さらに、IUTの場合 ”by Aschbacher and Smith was 1221 pages long”の場合よりも、その成否の影響が圧倒的に大きい ”by Aschbacher and Smith was 1221 pages long”なんて、自分に関係無いですむ人が大半でしょうけど
でも、IUTはそうではない だから問題なのです
(参考) https://en.wikipedia.org/wiki/List_of_long_mathematical_proofs List of long mathematical proofs
Long proofs The length of unusually long proofs has increased with time. As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof.
・2004 Quasithin groups The classification of the simple quasithin groups by Aschbacher and Smith was 1221 pages long, one of the longest single papers ever written. ・2004 Classification of finite simple groups. The proof of this is spread out over hundreds of journal articles which makes it hard to estimate its total length, which is probably around 10000 to 20000 pages.
>>・騒いでる Woitとか David Roberts とか >・彼らは査読プロセスを信用していない
David Robertsのことは知らないので スルーとして Woit氏は、「騒ぎ屋」さんですね(^^; かれ ”Woit, a critic of string theory, has published a book Not Even Wrong and writes a blog of the same name.[2]”です そして、American theoretical physicistであって、数学は単なる ” lecturer ”です
(参考) https://en.wikipedia.org/wiki/Peter_Woit Peter Woit Peter Woit (/?w??t/; born September 11, 1957) is an American theoretical physicist. He is a senior lecturer in the Mathematics department at Columbia University. Woit, a critic of string theory, has published a book Not Even Wrong and writes a blog of the same name.[2] (引用
そうですね では、下記など David Roberts氏 彼の研究分野は、どちらかと言えば、物理数学系ですね 数論や、まして、IUTの数学を論じる資格は ないってことですね
(参考) https://thehighergeometer.wordpress.com/ theHigherGeometer https://thehighergeometer.wordpress.com/about/ About My name is David Michael Roberts. I am currently a research associate at the Institute for Geometry and its Applications in the School of Mathematical Sciences at the University of Adelaide. You can see a recent copy of my CV here. https://thehighergeometer.files.wordpress.com/2020/03/david_roberts_cv_mar_2020.pdf David Michael ROBERTS March 20, 2020
M Flax @m_flax 返信先:@MugaShohouさん Well, Roberts *is* an expert in category theory and mathematical formalisms. So his feedback as to how the presentation can be made easier to verify shouldn't be dismissed as a "non-expert".
https://twitter.com/m_flax/status/1258339169694334977 (deleted an unsolicit
David Roberts 氏について Peter Scholze says:で、 ”I highly doubt your sentiment that the possibility of doing mistakes is not correlated with how well your language is adapted to the mathematics at hand.” とたしなめられていますね
まあ、私は David Roberts 氏も、”sentiment”に騒ぐお方とみましたけどねw(^^; それはともかく、今しばし見ていれば、世界の遠アーベルや、IUTに近い数論専門家が 動き出すと見ています ( David Roberts 氏 のような ”sentiment”の発言で、当方が 騒ぐ必要なしと思いますけど)
(参考) https://www.math.columbia.edu/~woit/wordpress/?p=11709#comments (woitブログ) Not Even Wrong Latest on abc Posted on April 3, 2020 by woit
Peter Scholze says: April 15, 2020 at 4:31 pm
Finally a short answer to David Roberts’ last message: I highly doubt your sentiment that the possibility of doing mistakes is not correlated with how well your language is adapted to the mathematics at hand.
これ 違うな 超弦理論がある Woit氏が” a critic of string theory, has published a book Not Even Wrong”(>>878)と批判するのは、物理の観測に結び付かないから、批判しているのです でも、超弦理論自身は、だれも数学とは見ない。あくまで物理です。観測に基づかない物理の理論です
(参考) https://ja.wikipedia.org/wiki/%E3%83%A4%E3%83%B3%E2%80%93%E3%83%9F%E3%83%AB%E3%82%BA%E6%96%B9%E7%A8%8B%E5%BC%8F%E3%81%A8%E8%B3%AA%E9%87%8F%E3%82%AE%E3%83%A3%E3%83%83%E3%83%97%E5%95%8F%E9%A1%8C ヤン?ミルズ方程式と質量ギャップ問題 ヤン?ミルズ方程式の存在と質量ギャップ問題(ヤン?ミルズほうていしきのそんざいとしつりょうぎゃっぷもんだい、英: Yang?Mills existence and mass gap)とは、量子色力学および数学上の未解決問題である。2000年、アメリカ合衆国のクレイ数学研究所はミレニアム懸賞問題の一つとしてこの問題に100万ドルの懸賞金をかけた。 公式な問題記述 問題文は次の通り[1]。 ヤン・ミルズ方程式の存在と質量ギャップ問題。任意のコンパクトな単純ゲージ
「集合 X から生成された自由群を F とし、 R を X 上の語からなる集合とする。 このとき R の正規閉包 N による商群を G = F/N
1064 名前: とおく。 これをG=<X|R>と表し、(生成元と基本関係による)群の表示という。
またこのとき、X の元を生成元、R の元を関係(または定義関係、基本関係) といい、 群 G は生成元と基本関係によって与えられると言う。 基本関係 w ∈ R に対し、式 w = 1 (1 は G の単位元) は基本関係式とも呼ばれる。 略式の言い方をすれば、N で割ることは G が自由群 F の元のうち、 R に属する元を単位元 1 に等しいものとみなして得られるものであること を意味している。
X が有限集合であるとき G は有限生成であるといい、 R が有限集合であるとき G は有限関係であるという。 また X と R が共に有限集合のとき、 群 G は有限型であるまたは有限表示されるという。」