A Finite Order Arithmetic Foundation for Cohomology

Large-structure tools like toposes and derived categories in cohomology never go far from arithmetic in practice, yet existing formalizations are stronger than ZFC. We formalize the practical insight by grounding the entire toolkit of EGA and SGA at the level of finite order arithmetic. Grothendieck pre-empted many set theoretic issues in cohomology by positing a universe: “a set ‘large enough’ that the habitual operations of set theory do not go outside it” (SGA 1 VI.1 p. 146). His universes prove Zermelo-Fraenkel set theory with choice (ZFC) is consistent, so ZFC cannot prove they exist. This paper founds the EGA and SGA on axioms with the proof theoretic strength of finite order arithmetic. Even Zermelo set theory (Z) proves these axioms consistent. So functors from all modules on a scheme to all Abelian groups achieve unity by generality indeed, but with little generality in the set theoretic sense. Outline. Section 1 introduces the issues by formalizing EGA and SGA with weaker universes within ZFC. Section 2 opens the serious reduction to Mac Lane set theory (MC) a fragment of ZFC with the strength of finite order arithmetic. Section 3 develops category theory in MC, with special care for injective resolutions. Section 4 gives a conservative extension of MC with classes and collections of classes so Sections 5–6 can develop large-scale structures of cohomology at the strength of finite order arithmetic. We use a simple notion of U-category which Grothendieck rejected at SGA 4 I.1.2 (p. 5). We cannot go through all the SGA and EGA. Most of that is commutative algebra elementary in logical strength. We focus on cohomology, geometric morphisms, duality and derived categories, and fibred categories. This supports the entire EGA and SGA. Section 7 relates this to prospects for proving Fermat’s Last Theorem (FLT) in Peano Arithmetic. 1. Replacement, separation, and the quick route to ZFC We can formalize EGA and SGA verbatim within ZFC by using a weaker definition of universe. We only modify the proof that cohomology groups exist. Zermelo set theory with choice (ZC) is ZFC without foundation or replacement but with the separation axiom scheme. This says for any set A and formula Ψ(x)