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Colin McLarty,The Large structures of Grothendieck founded on finite-order arithmetic. Rev.Symbol.Logo
＜AI による概要＞
Colin McLarty's paper, "The Large Structures of Grothendieck Founded on Finite-Order Arithmetic," published in the Review of Symbolic Logic in 2020, argues that the complex tools of Grothendieck-style algebraic geometry, such as toposes and derived categories, can be founded on the much weaker system of finite-order arithmetic, rather than the stronger ZFC set theory typically used for such foundations. This research aims to close the foundational gap by showing that these powerful "large structure" tools only require a significantly weaker logical strength to be established, with one specific topos of sets already requiring this level of strength.
・Core argument: McLarty demonstrates that the theorems of Grothendieck's Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique du Bois-Marie (SGA), along with derived categories, can be formally grounded using finite-order arithmetic.
・Weakest possible foundation: The paper establishes that finite-order arithmetic is the weakest possible foundation for these tools because even a single elementary topos of sets with infinity is already this strong.
・Implication for set theory: This finding implies that one does not need the full strength of ZFC, which is stronger than finite-order arithmetic, to prove the consistency of these large structures. The work shows that their foundations can be established with axioms that have the same proof-theoretic strength as finite-order arithmetic, which Zermelo set theory (Z) proves is consistent.
・Practical insight: The work formalizes the practical insight that tools of cohomology, while theoretically requiring large structures, often stay close to arithmetic in their actual application.
・Publication details: The paper was published in the Review of Symbolic Logic (Volume 13, Issue 2, 2020), with the doi: 10.1017/s1755020319000340 and can be found on arXiv as paper 1102.1773.
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