Inter-universal geometry とABC 予想55

[過去ﾛｸﾞ] Inter-universal geometry とABC 予想55 (1002ﾚｽ)
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316: １３２人目の素数さん [] 2024/04/29(月) 23:51:39.15 ID:Dprx2Ixj

>>294
>モチベーションは、ABC予想を解きたい
>本質の課題をつきつめると、属性方程式を解きたい。
>ところが、基礎の公理により通常の集合論ではあり得ないので、集合論を拡大したい。

・発想はその通りで 下記IUTのIのP102にあるが、[IUTchIV]Remark 3.3.1(i)を見ろという
・[IUTchIV], Remark 3.3.1(i)では、「種」の概念で”基礎の公理に違反することなく「∈ ループをシミュレートする」こと（が可能）です”（下記）とある
大山鳴動して鼠一匹

(参考)
https://www.kurims.kyoto-u.ac.jp/~motizuki/papers-japanese.html
望月新一
https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf
[1] Inter-universal Teichmuller Theory I: Construction of Hodge Theaters 200518
P102
If one thinks of the relation “∈” between sets in axiomatic set theory as determining
a “tree”, then
the point of view of (b) is reminiscent of the point of view of [IUTchIV],§3,
where one is concerned with constructing some sort of artiﬁcial solution to
the “membership equation a ∈ a” [cf. the discussion of [IUTchIV], Remark 3.3.1(i)].

https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf
[4] Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations 200422
P75
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.
略
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.
(google訳)
つまり
「種」の概念により、たとえば、検討中の集合論の ∈ 構造の異なるレベルで発生する種オブジェクト間の種同型写像を考慮することができます。
— つまり、大まかに言えば、基礎の公理に違反することなく「∈ ループをシミュレートする」こと（が可能）です。

http://rio2016.5ch.net/test/read.cgi/math/1712989377/316

435: １３２人目の素数さん [] 2024/05/02(木) 13:22:54.46 ID:D4jdpvN5

>>423
>基礎論は破ってるよ

基礎論破ってない と本人が言っている
　>>316 再録
"artiﬁcial solution to the "membership equation a ∈ a""
については、[cf. the discussion of [IUTchIV], Remark 3.3.1(i)]で
"大まかに言えば、基礎の公理に違反することなく「∈ ループをシミュレートする」こと（が可能）です"
が結論

(参考)
https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf
Inter-universal Teichmuller TheoryI
P102
(b) an isomorphism, or identiﬁcation, between v [i.e., a prime of F ] and
v'[i.e., a prime of K] which [manifestly — cf., e.g., [NSW], Theorem
12.2.5] fails to extend to an isomorphism between the respective prime decomposition trees over v and v'.

If one thinks of the relation “∈” between sets in axiomatic set theory as determining a "tree", then
the point of view of (b) is reminiscent of the point of view of [IUTchIV],§3,
where one is concerned with constructing some sort of artiﬁcial solution to
the “membership equation a ∈ a” [cf. the discussion of [IUTchIV], Remark 3.3.1(i)].

https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf
Inter-universal Teichmuller TheoryIV
P75
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.
略
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.
(google訳)
「種」の概念により、たとえば、検討中の集合論の ∈ 構造の異なるレベルで発生する種オブジェクト間の種同型写像を考慮することができます
— つまり、大まかに言えば、基礎の公理に違反することなく「∈ ループをシミュレートする」こと（が可能）です

http://rio2016.5ch.net/test/read.cgi/math/1712989377/435

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