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Grothendieck originally developed etale cohomology in an extremely general setting, working with concepts such as Grothendieck toposes and Grothendieck universes.
With hindsight, much of this machinery proved unnecessary for most practical applications of the etale theory, and Deligne (1977) gave a simplified exposition of etale cohomology theory. Grothendieck's use of these universes
(whose existence cannot be proved in ZFC) led to some uninformed speculation that etale cohomology and its applications (such as the proof of Fermat's last theorem) needed axioms beyond ZFC.
In practice etale cohomology is used mainly for constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with a little care it can be constructed in this case without using any uncountable sets, and this can easily be done in ZFC (and even in much weaker theories).

Etale cohomology quickly found other applications, for example Deligne and Lusztig used it to construct representations of finite groups of Lie type; see Deligne?Lusztig theory.
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