[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 77 (1002ﾚｽ)
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250(3): 現代数学の系譜 雑談 ◆yH25M02vWFhP 11/05(水)10:13 ID:K/Lr81ky(2/12) AAS
つづき
（追加）>>247より
外部ﾘﾝｸ:mathoverflow.net
How strong a set theory is necessary for practical purposes in sheaf theory?
asked May 14, 2020 user158035
Is it known how much of ZFC is actually necessary for the basic, familiar constructions and theorems in sheaf theory, along the lines of section II.1 (and its exercises) in Hartshorne's "Algebraic Geometry" textbook?
1 Answer
Colin McLarty has looked into this
The large structures of Grothendieck founded on finite order arithmetic, Review of Symbolic Logic 13 issue 2 (2020) pp. 296--325, doi:10.1017/S1755020319000340, 外部ﾘﾝｸ[1773]:arxiv.org
with abstract (emphasis added):
省5
255(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 11/05(水)11:39 ID:K/Lr81ky(6/12) AAS
>>250
>Colin McLarty has looked into this
>The large structures of Grothendieck founded on finite order arithmetic, Review of Symbolic Logic 13 issue 2 (2020) pp. 296--325, doi:10.1017/S1755020319000340, arxiv.org/abs/1102.1773
これ、リンクのarxivは 2014年版だな
30 Apr 2014 COLIN MCLARTY
1. Outline
Finite order arithmetic (Takeuti, 1987, Part II), or simple type theory with infinity, is n-th order arithmetic for all finite n. It deals with numbers, sets of numbers, and sets of those, up through any fixed finite level. Sections 2– 3 develop basic cohomology in any one of several set theories equivalent to this.
Sections 4–5 give a weak notion of a universe U, and a simpler notion of Ucategory than Grothendieck’s (SGA 4 I.1.2), in a theory of classes and collections conservative over set theory. Section 6 proves standard theorems on toposes, derived categories, and fibered categories. This is the weakest possible level for Grothendieck’s tools since a single elementary topos of sets with infinity is already as strong as finite order arithmetic.
Section 7 relates this to proofs of Fermat’s Last Theorem.
つづく
313(11): 現代数学の系譜 雑談 ◆yH25M02vWFhP 11/05(水)23:27 ID:BZV1IQOW(4/5) AAS
>>306
(引用開始)
セタへの問題
>Pを集合論の論理式, Pのz,x以外の自由変数はw_0,,w_i-1で, yはPの自由変数でないとする.
のとき
>∀w_0…∀w_i-1∀x∃y∀z(z∈y⇔z∈x⋀P(z))
の自由変数をすべて書け
(引用終り)
ふっふ、ほっほ
あのな、それよりか
省10
760(4): 現代数学の系譜 雑談 ◆yH25M02vWFhP 11/11(火)12:53 ID:3Cq8ZFbO(1/4) AAS
当事者です （＾＾
>>313より再録
ID:BZV1IQOW
>>306
(引用開始)
セタへの問題
>Pを集合論の論理式, Pのz,x以外の自由変数はw_0,,w_i-1で, yはPの自由変数でないとする.
のとき
>∀w_0…∀w_i-1∀x∃y∀z(z∈y⇔z∈x⋀P(z))
の自由変数をすべて書け
省31
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