つづき （参考） 関連： 望月新一（数理研） www.kurims.kyoto-u.ac.jp/~motizuki/ News - Ivan Fesenko https://www.maths.nottingham.ac.uk/plp/pmzibf/nov.html Explicit estimates in inter-universal Teichmuller theory, by S. Mochizuki, I. Fesenko, Y. Hoshi, A. Minamide, W. Porowski, RIMS preprint in November 2020, updated in June 2021, accepted for publication in September 2021 https://ivanfesenko.org/wp-content/uploads/2021/11/Explicit-estimates-in-IUT.pdf NEW!! (2020-11-30) いわゆる南出論文 より P4 Theorem A. (Effective versions of ABC/Szpiro inequalities over mono-complex number
9 名前：fields) Theorem B. (Effective version of a conjecture of Szpiro) Corollary C. (Application to “Fermat’s Last Theorem”) P56 Corollary 5.9. (Application to a generalized version of “Fermat’s Last Theorem”) Let l, m, n be positive integers such that min{l, m, n} > max{2.453 ・ 10^30, log2 ||rst||C, 10 + 5 log2(rad(rst))}. Then there does not exist any triple (x, y, z) ∈ S of coprime [i.e., the set of prime numbers which divide x, y, and z is empty] integers that satisfies the equation
Theorem Bで、Effective ”Szpiro”も出る 但し、”effective versions of the Vojta”への言及がないので、Vojtaは 未だみたい ここ、一山当てる狙い目かもねｗ 他に、IUT関連 ・[R8] Higher adelic theory, talk at the Como School, September 2021 https://ivanfesenko.org/wp-content/uploads/2021/10/hat.pdf ・[R7] IUT and modern number theory, talk at the RIMS workshop on IUT Summit, September 2021 https://ivanfesenko.org/wp-content/uploads/2021/10/mntiut.pdf ? [R5] Class field theory, its three main generalisations, and applications, May 2021, EMS Surveys 8(2021) 107-133 https://ivanfesenko.org/wp-content/uploads/2021/10/232.pdf
Yourpedia 宇宙際タイヒミュラー理論 （URLが通らないので検索たのむ） https://ja.wikipedia.org/wiki/%E5%AE%87%E5%AE%99%E9%9A%9B%E3%82%BF%E3%82%A4%E3%83%92%E3%83%9F%E3%83%A5%E3%83%A9%E3%83%BC%E7%90%86%E8%AB%96 宇宙際タイヒミュラー理論 Wikipedia https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCller_theory 英Inter-universal Teichmuller theory 英 Wikipedia https://ja.wikipedia.org/wiki/ABC%E4%BA%88%E6%83%B3 ABC予想 https://en.wikipedia.org/wiki/Abc_conjecture 英abc conjecture
https://www.math.arizona.edu/~kirti/ から Recent Research へ入る Kirti Joshi Recent Research論文集 新論文（IUTに着想を得た新理論） https://arxiv.org/pdf/2106.11452.pdf Construction of Arithmetic Teichmuller Spaces and some applications Preliminary version for comments Kirti Joshi June 23, 2021
https://www.uvm.edu/~tdupuy/papers.html [ Taylor Dupuy's Homepage] 論文集 なお、（メモ）TAYLOR DUPUYは、arxiv投稿で [SS17]を潰した（下記） https://arxiv.org/pdf/2004.13108.pdf PROBABILISTIC SZPIRO, BABY SZPIRO, AND EXPLICIT SZPIRO FROM MOCHIZUKI’S COROLLARY 3.12 TAYLOR DUPUY AND ANTON HILADO Date: April 30, 2020. P14 Remark 3.8.3. (1) The assertion of [SS17, pg 10] is that (3.3) is the only relation between the q-pilot and Θ-pilot degrees. The assertion of [Moc18, C14] is that [SS17, pg 10] is not what occurs in [Moc15a]. The reasoning of [SS17, pg 10] is something like what follows:
P15 (2) We would like to point out that the diagram on page 10 of [SS17] is very similar to the diagram on §8.4 part 7, page 76 of the unpublished manuscript [Tan18] which Scholze and Stix were reading while preparing [SS17]. References [SS17] Peter Scholze and Jakob Stix, Why abc is still a conjecture., 2017. 1, 1, 1e, 2, 7.5.3 ( https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf Date: July 16, 2018. https://ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf Date: August 23, 2018. ) [Tan18] Fucheng Tan, Note on IUT, 2018. 1, 2
なお "[SS17] Peter Scholze and Jakob Stix, Why abc is still a conjecture., 2017."は、2018の気がする ”[Tan18] Fucheng Tan, Note on IUT, 2018. 1, 2”が見つからない。”the unpublished manuscript [Tan18]”とはあるのだが（＾＾ 代わりに、ヒットした下記でも、どぞ （2018の何月かが不明だが、2018.3のSS以降かも）
www.kurims.kyoto-u.ac.jp/~motizuki/Tan%20---%20Introduction%20to%20inter-universal%20Teichmuller%20theory%20(slides).pdf Introduction to Inter-universal Teichm¨uller theory Fucheng Tan RIMS, Kyoto University 2018 To my limited experiences, the following seem to be an option for people who wish to get to know IUT without spending too much time on all the details. ・ Regard the anabelian results and the general theory of Frobenioids as blackbox. ・ Proceed to read Sections 1, 2 of [EtTh], which is the basis of IUT. ・ Read [IUT-I] and [IUT-II] (briefly), so as to know the basic definitions. ・ Read [IUT-III] carefully. To make sense of the various definitions/constructions in the second half of [IUT-III], one needs all the previous definitions/results. ・ The results in [IUT-IV] were in fact discovered first. Section 1 of [IUT-IV] allows one to see the construction in [IUT-III] in a rather concrete way, hence can be read together with [IUT-III], or even before. S. Mochizuki, The ´etale theta function and its Frobenioid-theoretic manifestations. S. Mochizuki, Inter-universal Teichm¨uller Theory I, II, III, IV.
www.kurims.kyoto-u.ac.jp/daigakuin/Tan.pdf 教員名: 譚 福成（Tan, Fucheng） P-adic Hodge theory plays an essential role in Mochizuki's proof of Grothendieck's Anabelian Conjecture. Recently, I have been studying anabeian geometry and Mochizuki's Inter-universal Teichmuller theory, which is in certain sense a global simulation of p-adic comparison theorem.
https://www.maths.nottingham.ac.uk/plp/pmzibf/mp.html Ivan Fesenko - Research in texts https://www.maths.nottingham.ac.uk/plp/pmzibf/232.pdf [R5] Class field theory, its three main generalisations, and applications pdf, May 2021
P16の後半に面白い図がある
コピーペースト下記 Here are some relations between the three generalisations of CFT and their further developments:
2dLC?－－ 2dAAG－－－ IUT l ／ ｜ ｜ l ／ ｜ ｜ l／ ｜ ｜ LC 2dCFT anabelian geometry ＼ ｜ ／ ＼ ｜ ／ ＼ ｜ ／ CFT 注）記号： Class Field Theory (CFT), Langlands correspondences (LC), 2dAAG = 2d adelic analysis and geometry, two-dimensional (2d) (P8 "These generalisations use fundamental groups: the etale fundamental group in anabelian geometry, representations of the etale fundamental group (thus, forgetting something very essential about the full fundamental group) in Langlands correspondences and the (abelian) motivic A1 fundamental group (i.e. Milnor K2) in two-dimensional (2d) higher class field theory.") https://www.kurims.kyoto-u.ac.jp/~motizuki/ExpHorizIUT21/WS4/documents/Fesenko%20-%20IUT%20and%20modern%20number%20theory.pdf Fesenko IUT and modern number theory つづく
（IUTに対する批判的レビュー） https://zbmath.org/07317908 https://zbmath.org/pdf/07317908.pdf Mochizuki, Shinichi Inter-universal Teichmuller theory. I: Construction of Hodge theaters. (English) Zbl 07317908 Publ. Res. Inst. Math. Sci. 57, No. 1-2, 3-207 (2021). Reviewer: Peter Scholze (Bonn)
BuzzardのICM22講演原稿 Inter-universal geometry とABC 予想47 https://rio2016.5ch.net/test/read.cgi/math/1635332056/84 84 名前：38[] 投稿日：2021/12/23(木) 19:42:33.42 ID:iz9G4jw+ [1/2] Buzzardの原稿が出たヨ！ https://arxiv.org/abs/2112.11598 >A great example is Mochizuki’s claimed proof of the ABC conjecture [Moc21]. >This proof has now been published in a serious research journal, however >it is clear that it is not accepted by the mathematical community in general.
86 名前：１３２人目の素数さん[] 投稿日：2021/12/23(木) 20:46:56.21 ID:a0F2ZqKI >>84 ホントに出ていたね。その引用部分の少し後に次のことが書かれている。 Furthermore, the key sticking point right now is that the unbelievers argue that more details are needed in the proof of Corollary 3.12 in the main paper, and the state of the art right now is simply that one cannot begin to formalise this corollary without access to these details in some form (for example a paper proof containing far more information about the argument) (引用終り)
”Comments: 28 pages, companion paper to ICM 2022 talk”と明記もあるね 思うに、その意図は、「反論あるなら言ってきてね。反論の機会を与える。反論なき場合はこのまま総会発表とする」ってことか （西洋流で、「黙っていたから 認めたってことじゃん」みたいなｗ） 普通は、こんな形でプレプリ出さない気がするな さあ、面白くなってきたかも ドンパチ派手にやってほしい
>>34 追加 https://ja.wikipedia.org/wiki/%E9%81%BA%E4%BC%9D%E7%9A%84%E6%9C%89%E9%99%90%E9%9B%86%E5%90%88 遺伝的有限集合 https://en.wikipedia.org/wiki/Hereditarily_finite_set Hereditarily finite set Representation This class of sets is naturally ranked by the number of bracket pairs necessary to represent the sets: ・{} (i.e.Φ, the Neumann ordinal "0"), ・{{}} (i.e. {Φ} or {0}, the Neumann ordinal "1"), ・{{{}}}, ・{{{{}}}} and then also {{},{{}}} (i.e. {0,1}, the Neumann ordinal "2"), ・{{{{{}}}}}, {{{},{{}}}} as well as {{},{{{}}}}, ・... sets represented with 6 bracket pairs, e.g. {{{{{{}}}}}}, ・... sets represented with 7 bracket pairs, e.g. {{{{{{{}}}}}}}, ・... sets represented with 8 bracket pairs, e.g. {{{{{{{{}}}}}}}} or {{},{{}},{{},{{}}}} (i.e. {0,1,2}, the Neumann ordinal "3") ... etc. In this way, the number of sets with n}n bracket pairs is[1] 1,1,1,2,3,6,12,25,52,113,247,548,1226,2770,6299,14426,・・・ Axiomatizations ZF The hereditarily finite sets are a subclass of the Von Neumann universe. Here, the class of all well-founded hereditarily finite sets is denoted Vω. Note that this is also a set in this context. If we denote by p(S) the power set of S, and by V0 the empty set, then Vω can be obtained by setting V1 = p(V0), V2 = p(V1),..., Vk = p(Vk-1),... and so on. Thus, Vω can be expressed as Vω=∪ k=0～∞ Vk. We see, again, that there are only countably many hereditarily finite sets: Vn is finite for any finite n, its cardinality is n-12 (see tetration), and the union of countably many finite sets is countable. (引用終り)
遺伝的有限集合、Hereditarily finite set 「naturally ranked by the number of bracket pairs」で そのbracket（カッコ）の深さのシングルトン達、例えば深さ6 with 6 bracket pairs, e.g. {{{{{{}}}}}}とか出てくるよ そして、ZFでは、Vω=∪ k=0～∞ Vk だ だから、ω重シングルトン あるんじゃね？
https://en.wikipedia.org/wiki/Hereditarily_finite_set Hereditarily finite set
Discussion A symbol for the class of hereditarily finite sets is H_ｱﾚﾌ0, standing for the cardinality of each of its member being smaller than ｱﾚﾌ0. Whether H_ｱﾚﾌ0 is a set and statements about cardinality depend on the theory in context.
Axiomatizations Theories of finite sets The set Φ also represents the first von Neumann ordinal number, denoted 0. And indeed all finite von Neumann ordinals are in ｱﾚﾌ0 and thus the class of sets representing the natural numbers, i.e it includes each element in the standard model of natural numbers. Robinson arithmetic can already be interpreted in ST, the very small sub-theory of Z^{-} with axioms given by Extensionality, Empty Set and Adjunction.
Indeed, ｱﾚﾌ0 has a constructive axiomatizations involving these axiom and e.g. Set induction and Replacement.
Their models then also fulfill the axioms consisting of the axioms of Zermelo?Fraenkel set theory without the axiom of infinity. In this context, the negation of the axiom of infinity may be added, thus proving that the axiom of infinity is not a consequence of the other axioms of set theory.
https://en.wikipedia.org/wiki/Von_Neumann_universe Von Neumann universe Finite and low cardinality stages of the hierarchy The set Vω has the same cardinality as ω. The set Vω+1 has the same cardinality as the set of real numbers. (引用終り)
これで １．Von Neumann universeで、”The set Vω has the same cardinality as ω. ”つまり、Vωの濃度はωで、可算無限 ２，だから、Vω=∪ k=0～∞ Vkで、k=0～∞は、ちゃんと∞まで渡るよ ３．なお、上記 Hereditarily finite setの”Discussion”に書いてあるように、Axiomatizations Theories of finite sets と ZFでは、扱いが違うってことです
>１．Von Neumann universeで、”The set Vω has the same cardinality as ω. ”つまり、Vωの濃度はωで、可算無限 >２，だから、Vω=∪ k=0～∞ Vkで、k=0～∞は、ちゃんと∞まで渡るよ ∪[k∈N]{k}=Nの濃度もωですよ？ 濃度がωだからといって∞なるワケノワカラナイモノが必要とは言えませんよ？
数学における存在とは？ 下記Von Neumann universe を例に ”Since existence is a difficult concept, one typically replaces the existence question with the consistency question, that is, whether the concept is free of contradictions. A major obstacle is posed by Godel's incompleteness theorems, which effectively imply the impossibility of proving the consistency of ZF set theory in ZF set theory itself, provided that it is in fact consistent.[12] The integrity of the von Neumann universe depends fundamentally on the integrity of the ordinal numbers, which act as the rank parameter in the construction, and the integrity of transfinite induction, by which both the ordinal numbers and the von Neumann universe are constructed. ”
(参考) https://en.wikipedia.org/wiki/Von_Neumann_universe Von Neumann universe Definition Finite and low cardinality stages of the hierarchy The set Vω has the same cardinality as ω. The set Vω+1 has the same cardinality as the set of real numbers.
The existential status of V Since the class V may be considered to be the arena for most of mathematics, it is important to establish that it "exists" in some sense. Since existence is a difficult concept, one typically replaces the existence question with the consistency question, that is, whether the concept is free of contradictions. A major obstacle is posed by Godel's incompleteness theorems, which effectively imply the impossibility of proving the consistency of ZF set theory in ZF set theory itself, provided that it is in fact consistent.[12]
The integrity of the von Neumann universe depends fundamentally on the integrity of the ordinal numbers, which act as the rank parameter in the construction, and the integrity of transfinite induction, by which both the ordinal numbers and the von Neumann universe are constructed. The integrity of the ordinal number construction may be said to rest upon von Neumann's 1923 and 1928 papers.[13] The integrity of the construction of V by transfinite induction may be said to have then been established in Zermelo's 1930 paper.[7] (引用終り) 以上
(参考) https://en.wikipedia.org/wiki/Zermelo_set_theory Zermelo set theory (sometimes denoted by Z-) Contents 1 The axioms of Zermelo set theory 2 Connection with standard set theory 3 Mac Lane set theory 4 The aim of Zermelo's paper 5 The axiom of separation 6 Cantor's theorem 7 See also (引用終り) 以上
>>38 補足 >遺伝的有限集合、Hereditarily finite set >「naturally ranked by the number of bracket pairs」で >そのbracket（カッコ）の深さのシングルトン達、例えば深さ6 with 6 bracket pairs, e.g. {{{{{{}}}}}}とか出てくるよ >そして、ZFでは、Vω=∪ k=0～∞ Vk だ >だから、ω重シングルトン あるんじゃね？
文献を誤読している人がいるので補足する （>>38 ＆>>46再録） https://en.wikipedia.org/wiki/Hereditarily_finite_set Hereditarily finite set
Discussion A symbol for the class of hereditarily finite sets is H_ｱﾚﾌ0, standing for the cardinality of each of its member being smaller than ｱﾚﾌ0. Whether H_ｱﾚﾌ0 is a set and statements about cardinality depend on the theory in context.
Axiomatizations Theories of finite sets The set Φ also represents the first von Neumann ordinal number, denoted 0. And indeed all finite von Neumann ordinals are in ｱﾚﾌ0 and thus the class of sets representing the natural numbers, i.e it includes each element in the standard model of natural numbers. Robinson arithmetic can already be interpreted in ST, the very small sub-theory of Z^{-} with axioms given by Extensionality, Empty Set and Adjunction.
Their models then also fulfill the axioms consisting of the axioms of Zermelo-Fraenkel set theory without the axiom of infinity. In this context, the negation of the axiom of infinity may be added, thus proving that the axiom of infinity is not a consequence of the other axioms of set theory.
ZF The hereditarily finite sets are a subclass of the Von Neumann universe. Here, the class of all well-founded hereditarily finite sets is denoted Vω. Note that this is also a set in this context.
If we denote by p(S) the power set of S, and by V0 the empty set, then Vω can be obtained by setting V1 = p(V0), V2 = p(V1),..., Vk = p(Vk-1),... and so on. Thus, Vω can be expressed as Vω=∪ k=0～∞ Vk. We see, again, that there are only countably many hereditarily finite sets: Vn is finite for any finite n, its cardinality is n-12 (see tetration)
92 名前：, and the union of countably many finite sets is countable. (引用終り)
１．書かれているように、Hereditarily finite setは、”cardinality depend on the theory in context”ってことです ２．つまり、”Theories of finite sets”＝有限集合理論 では、 例えば、”In this context, the negation of the axiom of infinity may be added” とあるように、無限公理の否定をあえて追加する議論もありってこと。この場合は当然、無限集合は否定されるってこと ３．で、ZFのcontextでは、無限公理は認める立場だ この立場は、古代ギリシャのユークリッドが、素数の無碍を証明したのと同じ（標準的立場） つまり、自然数集合N 元 1,2,3,・・ で、濃度ｱﾚﾌ0 つまり、1,2,3,・・ は可算無限個あるが、但し∀nたちは有限 よって、このcontextでは 上記の”Vω=∪ k=0～∞ Vk”も、正当化できる ４．「1,2,3,・・ は可算無限個あるが、但し∀nたちは有限」という この一見矛盾した状況が理解できないレベルならば、 カントールの順序数論では、まっとうに議論できるレベルじゃない（低レベル）ってことです 以上 []
>>84 追加 >the very small sub-theory of Z^{-} with axioms given by Extensionality, Empty Set and Adjunction.
さらに、ここ原文には、Z^{-}にはリンクが張ってあって、>>77の https://en.wikipedia.org/wiki/Zermelo_set_theory Zermelo set theory (sometimes denoted by Z-) にとぶ
で、誤解なきよう念のために書くが ”the very small sub-theory of Z^{-}”なので、 上記は Zermelo set theoryの一部ってことですよ
で、ついでに書くと 上記 Zermelo set theoryでは、 ”AXIOM VII. Axiom of infinity (Axiom des Unendlichen) "There exists in the domain at least one set Z that contains the null set as an element and is so constituted that to each of its elements a there corresponds a further element of the form {a}, in other words, that with each of its elements a it also contains the corresponding set {a} as element."” とあって、無限公理（Axiom of infinity）で、集合Zが存在すると言葉で書かれている
この集合Zは、ドイツ語で数Zahlからで（下記）、自然数の意味ですね で、Zermeloは、シングルトン{a}を使って、自然数Nができると Axiom des Unendlichenを書いた これにいろいろ批判があることも、上記リンク内に書いてある （さらに付言すると、上記Zermeloでは、ω重シングルトン自身は使っていないのです。 その一歩手前で、有限シングルトンを全部集めて自然数Nができるという議論だ だけど、ω重シングルトンを否定しているわけでもない）
で、”Zermelo allowed for the existence of urelements that are not sets and contain no elements; these are now usually omitted from set theories.” なんて文もある おれも最後は、ω重シングルトンで”existence of urelements”かもしれないが、まだそこまで行ってないよね
参考 https://de.wikipedia.org/wiki/Zahl Zahl (引用終り) 以上
独語の自然数の.wikipedia.があったので、参考に貼っておく https://de.wikipedia.org/wiki/Nat%C3%BCrliche_Zahl Naturliche Zahl
Inhaltsverzeichnis 1 Bezeichnungskonventionen 2 Axiomatisierung 3 Von Neumanns Modell der naturlichen Zahlen 4 Die naturlichen Zahlen als Teilmenge der reellen Zahlen 5 Siehe auch 6 Literatur (引用終り) 以上
