<IUT国際会議 2つのシリーズ> 1. http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/IUT-schedule.html RIMS Promenade in Inter-Universal Teichmuller Theory Org.: Collas (RIMS); Debes, Fresse (Lille). The seminar takes place every two weeks on Thursday for 2 hours by Zoom 17:30-19:30, JP time (9:30-11:30, UK time; 10:30-12:30 FR time) ? we refer to the Programme for descriptions of the talks and associated references. http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/documents/RIMS-Lille%20-%20Promenade%20in%20Inter-Universal%20Teichm%C3%BCller%20Theory.pdf 0004132人目の素数さん2022/01/09(日) 14:02:57.44ID:LV2O1tR+https://www.kurims.kyoto-u.ac.jp/~motizuki/project-2021-japanese.html 宇宙際タイヒミューラー理論の拡がり (4回とも無事終了です) なお、東大の重鎮 Atsushi Shiho (Univ. Tokyo, Japan)先生 8月末〜9月初めの二つのIUT会議に出席したようです
つづく 0008132人目の素数さん2022/01/09(日) 14:04:59.73ID:LV2O1tR+ つづき (参考) 関連: 望月新一(数理研) http://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 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
http://www.kurims.kyoto-u.ac.jp/~motizuki/Essential%20Logical%20Structure%20of%20Inter-universal%20Teichmuller%20Theory.pdf <PRIMS出版記念論文> [9] On the Essential Logical Structure of Inter-universal Teichmuller Theory in Terms of Logical AND "∧"/ Logical OR "∨" Relations: Report on the Occasion of the Publication of the Four Main Papers on Inter-universal Teichmuller Theory. PDF NEW!! (2021-03-06)
新一の「心の一票」 - 楽天ブログ shinichi0329/ (URLが通らないので検索たのむ) math jin:(IUTT情報サイト)ツイッター math_jin (URLが通らないので検索たのむ)
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)
http://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.
http://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.
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 0014132人目の素数さん2022/01/09(日) 14:07:10.96ID:LV2O1tR+ つづき
(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
取り敢えずこんなところで(^^ 0015DiverCity (りんかい線)2022/01/09(日) 16:31:11.06 このスレは国立大学工学部卒を詐称する 工業高校中退の中卒の誇大妄想狂 SET Aが立てた 反数学アラシスレです
デジタルコピー貼り付けはやめたまえ 馬鹿がわけもわからずリコウぶっても恥かくだけだ もういいかげん💩塗れで現れるのはやめてくれ 悪臭が耐え難い 0037132人目の素数さん2022/01/10(月) 19:45:19.61ID:NpIRFDiP>>34 で?ω重シングルトンは間違いだと認めるの? 時枝不成立も間違いだと認めれば? 0038132人目の素数さん2022/01/10(月) 20:27:52.98ID:MGTx95Re>>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? 0039DiverCity (りんかい線)2022/01/10(月) 20:45:36.05>>38 >ZFでは、Vω=∪ k=0〜∞ Vk だ
はい、誤り ZFでは、Vω=∪ k∈N Vk だ
∞はNの要素じゃありませーん
SET Aって、ほんと🐎🦌だなw
>だから、ω重シングルトン あるんじゃね?
だから、ω重シングルトンありませーん
SET Aって、ほんと🐎🦌だなw 0040132人目の素数さん2022/01/10(月) 21:46:28.26ID:KC/ZM0+6 セタは自然数とωの違いも分からなければ無限大自然数との違いも分かりません
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 0047132人目の素数さん2022/01/11(火) 01:06:57.77ID:RsLvQDWl>>46 >ちゃんと∞まで渡るよ ∞とは何ですか?
>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? 0048132人目の素数さん2022/01/11(火) 01:55:43.53ID:xrQGvxzy うっかり>>23をコテハンで書き込んでしまい DiverCity (りんかい線)の正体であるとバレてしまった間抜けな長野県の成績Fおじさん 東京住みの人なら付けないベタな名前に哀愁を感じる 0049132人目の素数さん2022/01/11(火) 06:05:13.62ID:w8+vN4kT>>46 おいそこのゾンビ finite set の意味を答えて見せろ此のガキ ∞まで含めて何でω止まりなんだかも答えろやゴミ
しかも長年無収入で親の財を食い潰し、嘘連投を咎め詰られたら「ここは便所の落書きクソ喰らえ」と不遜な開き直り。 世の中を深く深く冒涜し過ぎだお前は、冒涜だ冒涜。 0050132人目の素数さん2022/01/11(火) 06:30:53.62ID:rxr3xgBa>>46 >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
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 0053132人目の素数さん2022/01/11(火) 08:29:24.23ID:FjO7HUzR>>51 >ツェルメロのシングルトンで、可算多重を否定したい人は、どうぞ証明を ωの前者が存在しないので、ツェルメロのシングルトンなるものを{x}と書いたときxには最外カッコが無い。よってxは集合でない。よってツェルメロのシングルトンなるものも集合でない。よってZF上に存在しない。 0054132人目の素数さん2022/01/11(火) 17:15:35.90ID:13tp60RC 星裕一郎先生のツイッター おんなの匂いがしないな 大丈夫か? もう40だよね?
(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 0078132人目の素数さん2022/01/13(木) 08:22:21.95ID:Q7w9Eui5>>77 >なので、最低下記でも読まないと >まともな、数学の議論にならんよね なんでどーでもいーコピペには熱心なのに、肝心のω重シングルトンはコピペせんの? ペテン師だから? 0079132人目の素数さん2022/01/13(木) 09:23:09.01ID:i4s8QaxG おサルは情報系修士ですらないでしょ 挙げてきた書籍名が匿名掲示板過去スレから拾い上げてきた的外れな一覧な上にタイトルを知っているだけで読んでもいないと予防線を張る間抜けさに爆笑した 0080132人目の素数さん2022/01/13(木) 09:56:25.37ID:i4s8QaxG 二セ科学批判力ル卜が多用する特異語 [改訂版]
1. ニ セ 科 学 | エ セ 科 学 | 疑 似 科 学 2. ト ン デ モ | 【ペ テ ン 師】 | デ マ | ウ ソ | 詭 弁 | 病 気 3. 信 者 | 信 奉 者 | 教 祖 | 信 じ る 4. 負 け を 認 め て 黙 れ 5. 自 殺 | 氏 ね 6. 自 分 が 本 当 の 被 害 者 7. 月 刊 ム ー | オ カ ル ト | 宇 宙 人 8. 理 研 | S T A P 細 胞 | 小 保 方 | オ ボ カ タ 9. 岡 崎 | 丘 裂 き | 生 物 多 様 性 10. h i s s i . o r g | ウ ィ キ ペ デ ィ ア (w i k i p e d i a) 11. 悪 魔 の 証 明 12. キ チ ガ イ | 統 合 失 調 0081132人目の素数さん2022/01/13(木) 10:01:46.97ID:0h1VRMgw>>79 どうもです スレ主です そうかもです 0082132人目の素数さん2022/01/13(木) 10:02:16.24ID:0h1VRMgw>>78 どうもです スレ主です
んなわけ有っか此のバーカ。お前もクソもミソも一緒にする奴か、見境ねぇ奴ばかりだなぁ? 0084132人目の素数さん2022/01/13(木) 10:42:58.95ID:0h1VRMgw>>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?
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), 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 0086132人目の素数さん2022/01/13(木) 11:18:45.65ID:0h1VRMgw>>84 追加 >the very small sub-theory of Z^{-} with axioms given by Extensionality, Empty Set and Adjunction.
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