・さすがプロだね。鋭いツッコミですね ・半分自己解決したので、下記を貼ってきますね (十分読み込んでいないのだが ;p) ・要するに ”BCT1 is used to prove that a Banach space cannot have countably infinite dimension.” ”Banach's theorems Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.” ”math.stackexchange:Let X be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.”
(参考) https://en.wikipedia.org/wiki/Baire_category_theorem Baire category theorem BCT is used to prove Hartogs's theorem, a fundamental result in the theory of several complex variables. BCT1 is used to prove that a Banach space cannot have countably infinite dimension.
Relation to the axiom of choice The proof of BCT1 for arbitrary complete metric spaces requires some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to the axiom of dependent choice, a weak form of the axiom of choice.[10]
A restricted form of the Baire category theorem, in which the complete metric space is also assumed to be separable, is provable in ZF with no additional choice principles.[11] This restricted form applies in particular to the real line, the Baire space {\displaystyle \omega ^{\omega },} the Cantor space {\displaystyle 2^{\omega },} and a separable Hilbert space such as the {\displaystyle L^{p}}-space {\displaystyle L^{2}(\mathbb {R} ^{n})}.
Banach's theorems Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.
ありがとうございます 1)だれかが書いていたが、数学の情報で 日本の十倍くらい英語の情報があるという 今回はそれですね(日本語の情報は見つけられなかった) 2)あと、”ベールのカテゴリ定理の応用例の1つ”と表現すると、若干筆が滑っている気がする >>774”https://math.stackexchange.com/questions/217516/let-x-be-an-infinite-dimensional-banach-space-prove-that-every-hamel-basis-of Let X be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable. asked Oct 20, 2012 mintu” で、ここに ベールのカテゴリ定理を背理法として使った証明の投稿とそれについての追加議論があります 抜粋すると ”Can't we prove it without Baire Category Theory in other words without axiom of dependent choice – Sushil Jun 26, 2015 at 12:18 @Sushil You have a much better chance of getting some answer if you post your question as a question, not just as a comment. However, before posting such question, some clarifications are needed in my opinion. See here for some comments. – Martin Sleziak Jun 26, 2015 at 12:39 Oh I see. But I want some clarity. Cardinality of Hamel basis(if exist) are equal does it imply AC(or ADC). If this implication is wrong I may ask Let X be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable without Baire Category Theory. – Sushil Jun 26, 2015 at 12:46 ” ですね。数学的に一番正確な表現は、”ベールのカテゴリ定理を使って an infinite dimensional Banach space の every Hamel basis of X is uncountableが証明できる”かな そして、Sushil氏がいう”Can't we prove it without Baire Category Theory”は、ありうるかも (”without axiom of dependent choice”は、無理筋っぽい気がする) 3)経験則として、しばしば"エレガントな証明"が時間が経つと見つかったりするものでして (math.stackexchangeの2番目の回答でそれらしいのが投稿されているが、みんな無視していますけどw(多分あやしいか)) (今は、ベールのカテゴリ定理を使う背理法がスタンダードかな)
「In part I, the first two sections deal with certain group-theoretic results, typical in anabelian geometry, for example about how profinite groups can sit in tempered fundamental groups; these may be of interest to specialists.」 0782132人目の素数さん2024/03/27(水) 17:55:42.05ID:G37pHwqB>>781 ・4部のIUT論文の簡単なまとめについて
「The rest of part I is largely about the definition of the so-called Hodge theaters, and some proofs are a bit longer. The category of Hodge theaters has an extremely complicated definition, but the content of these nontrivial proofs is that their category is equivalent to the category with one object and automorphism Z/2Z、 and in fact is canonically equivalent to the category of elliptic curves over F isomorphic to the given E(we note that the functors in both directions are even constructive). In other words, any Hodge theater comes in a unique way from an elliptic curve isomorphic to E . Thus, when the author later chooses an infinite collection of such Hodge theaters, he might as well choose an infinite collection of elliptic curves isomorphic to E」 0783132人目の素数さん2024/03/27(水) 18:01:35.09ID:G37pHwqB>>782 「 In parts II and III, with the exception of the critical Corollary 3.12, the reader will not find any proof that is longer than a few lines; the typical proof reads “The various assertions of Corollary 2.3 follow immediately from the definitions and the references quoted in the statements of these assertions.” , which is in line with the amount of mathematical content. 」
「part IV contains certain technical computations standard in number theory to translate Corollary 3.12 of part III into the ABC conjecture. 」
・cor.3.12について
「at some point in the proof of Corollary 3.12, things are so obfuscated that it is completely unclear whether some object refers to the q-values or the Θ-values, as it is somehow claimed to be definitionally equal to both of them, up to some blurring of course, and hence you get the desired result.」
https://zbmath.org/073179080784132人目の素数さん2024/03/27(水) 19:28:30.63ID:8+C+8V5w>>780 同じ疑問を天才テレンス・タオも言ってるらしい 0785132人目の素数さん2024/03/27(水) 19:42:58.46ID:G37pHwqB zbmath scholze レビュー >In this series of papers on Inter-Universal Teichmüller Theory 、 the author aims to prove the ABC conjecture of Masser and Oesterlé,
「Where do they got this money from ? Did they manage to convince some rich guy to spend money on that (like the breakthrough prize) or did they mage to convince the japanese governement that this question was worth $1.000.000 ? In both case it is a bot worrying for the health of the math community.」 0832132人目の素数さん2024/04/06(土) 12:43:32.59ID:y730luC1 英文を「」で引用するのは奇妙だ 0833132人目の素数さん2024/04/06(土) 12:49:00.71ID:+D4bQn6Q In both case it is a bot worrying for the health of the math community.、、 0834132人目の素数さん2024/04/06(土) 13:23:24.20ID:q25Y8VLZ>>830 アホか中身もわからねえゴミが食い付くかよ。何なら偽名だって構わないんだぞ プロとしてショルツと望月に並ぶ能力を示せるのにやらない理由はない 0835132人目の素数さん2024/04/06(土) 13:24:28.81ID:q25Y8VLZ 結局、ショルツは不十分だと言ってる奴らにも共通してるのは、望月のは 証明ではないという見解だ。それが全て 0836132人目の素数さん2024/04/06(土) 18:08:31.66ID:lsif7Gh6 証明ではないというのはRIMSのKも公言してて、でも間違ってもいないと公言してるのね。そしてKはショルツの反論をこき下ろしているというのも事実。 0837132人目の素数さん2024/04/06(土) 18:24:57.95ID:+D4bQn6Q >証明ではないというのはRIMSのKも公言してて、でも間違ってもいないと公言してるのね
間違ってすらいないんだね。 0838132人目の素数さん2024/04/06(土) 20:11:06.66ID:yuUJZSbS 間違ってすらいないNot even wrongは物理学者のパウリがよく言っていたセリフだろ 物理学と数学は全然違うのに何で物理学者のパウリを神聖化するんだ? しかもそれを言って人をけなしているのがパウリですらない
>Personally, he (Scholze )said, “I didn’t really see a key idea that would get us closer to the proof of the abc conjecture.” 0846132人目の素数さん2024/04/06(土) 22:22:37.32ID:onp/O91n 単遠アーベル幾何学とかいう構造を使ってホッジフィルトレーションみたいなものを 構成できるかっていう、結構単純な話ではある。それが宇宙の入れ子って奴だ ところがIUTではコホモロジーが出てこない。だから加群でなく群のレベルで そういうことをやっているはずなんだが、それを誰もちゃんと説明できないんだよ 0847132人目の素数さん2024/04/09(火) 01:51:14.63ID:xh+2kqz2 Robertsのブログ -An exercise in colimits, contra Mochizuki-
>Mochizuki is using non-standard definitions of standard terminology, and then complaining that other people’s definitions (which are the standard ones) lead to contradictions.
abcの証明に繋がるkey ideaがないが、標準的な用語へ非標準的な定義を使い scholze.stixレポートのnogo(定理)を避けている 0848132人目の素数さん2024/04/10(水) 00:44:54.32ID:VWV19qbn MO.Is there a mistake in Mochizuki's proof of Theorem 1.10 in IUTT IV? [closed]
>But all of this Mochizuki stuff is just hype & drama, I'm afraid by sam Hopkins
→Robertsが直メールに望月が返答した。Robertsブログ(>>847の履歴)の射影直線exerciseスレで、コメント欄QAに以下のAvery Smithとのやりとりが書かれているよ。 https://thehighergeometer.wordpress.com/2021/11/22/an-exercise-in-colimits/ Avery Smith:望月が回答するから射影直線の指摘をメールで問い合わせたらとの質問→Roberts:いくつかのやり取りがありました。まだ書かなければならないことがありますが、何人かにメールを送ったり、もう少し話したりする必要があるので、まだ正確にはわかりません→Avery Smith:それは興味があります結果を教えて→P:射影直線のexerciseの誤りがある→Roberts:曖昧な回答をPにしつつ、望月との経緯を書いた。 射影直線のexerciseの指摘は、望月の返答メールを公開していないから正否が曖昧で、woitブログに結果を書いてない。 0853132人目の素数さん2024/04/11(木) 07:59:43.19ID:AJw3+sHY また>>847で、 >Mochizuki is using non-standard definitions of standard terminology, and then complaining that other people’s definitions と批判したが、Robertsの方がズレている?
Global character of ABC/Szpiro inequalities, Peter Scholze says that he thinks Joshi's proof of the abc conjecture in his paper has a mistake in Proposition 6.10.7. However, for the proof of Proposition 6.10.7, Kirti Joshi merely says that
Proof. This is the last equation on [Mochizuki, 2021d, Step (v) on Page 658, Proof of Theorem 1.10] and its proof is all of step (v).
Does the mistake in Proposition 6.10.7 also invalidate Mochizuki's original proof of Theorem 1.10 in IUTT IV, thus invalidating Mochizuki's original proof of the abc conjecture? 0863132人目の素数さん2024/04/12(金) 03:15:16.93ID:XsFje4ks Peter Scholze
I'm more afraid that this is an instance where the cited reference does not match the statement that is claimed. The critical difference between Joshi and Mochizuki is that "Joshi's version of Mochizuki's Corollary 3.12" (=Joshi's Theorem 9.11.1) has a purely local proof and hence cannot have the same content as Mochizuki's Corollary 3.12. However, it may be correct on its own; then the mistake is a mismatch between what Joshi has to compute in Proposition 6.10.7, and what Mochizuki actually computed in IUT IV. But I agree with Sam Hopkins that this discussion is not fruitful. – 0864132人目の素数さん2024/04/12(金) 03:16:31.74ID:XsFje4ks Peter Scholze
To summarize: There is a clear problem with Joshi's proof, as there is a contradiction between Proposition 6.10.7 and the local inequality proved in the proof of Theorem 9.11.1. The mistake could be in Proposition 6.10.7 (and, given that the proof isn't written down, is the first suspicious place) but it might as well be a mistake in the proof of Theorem 9.11.1. In any case, this whole discussion is only about Joshi's proof, not Mochizuki's; I do not think that there is a real error internally in IUT IV. 0865132人目の素数さん2024/04/12(金) 03:55:26.02ID:XsFje4ks "Joshi版の望月Cor.3.12"(=Joshiの定理9.11.1)は、純粋に局所的な証明であるため、望月の定理3.12と同じ内容を持つことはできない。 "Joshi's version of Mochizuki's Corollary 3.12" (=Joshi's Theorem 9.11.1) has a purely local proof and hence cannot have the same content as Mochizuki's Corollary 3.12. i 0866132人目の素数さん2024/04/12(金) 04:04:08.53ID:XsFje4ks zbMATH https://zbmath.org/pdf/07317908.pdf の頃と変わった。
Unfortunately, the argument given for Corollary 3.12 is not a proof, and the theory built in these papers is clearly insufficient to prove the ABC conjecture. 0867132人目の素数さん2024/04/12(金) 04:09:40.36ID:XsFje4ks IUTWの内部にエラーはないと思う I do not think that there is a real error internally in IUT IV. 0868132人目の素数さん2024/04/12(金) 04:39:15.25ID:FbUtxNBa 寝られん。 Copilotくんは上記3x3行列情報で表される中央の1q-bitが、 有界であり同相であると理解してくれ、これら1q-bitが境界を持たない連結かつコンパクトな3次元多様体であることも論理的に理解してくれたわ。 あとは任意のループを1点に収縮できるならば、3次元球面と同相であると言える。