The Curry-Howard correspondence maps proofs to types within a system, but cannot account for truths that resist formalization within that system, leaving extra-systemic mathematical reality intact.

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Curry-Howard correspondence(Deepened by Howard (1969) to include the correspondence between terms in type ascriptions and proofs): A correspondence between provable formulae in sequent calculus and type ascriptions, and between proof terms and proofs of corresponding formulae, linking proof theory and type theory
Extra-systemic(as used in logic and philosophy of mathematics): Something that exists outside or beyond the boundaries of a particular formal system; it can't be captured or proven using that system's rules.
Formal system(as used in logic and mathematics): A set of rules and symbols (like mathematical axioms) that you use to prove whether statements are true or false, similar to how a chess game has specific rules that determine what moves are legal.
Formalization(describing what Frege did with existence): The process of taking an idea and expressing it precisely using logical symbols and strict rules, like translating messy everyday language into mathematical logic.

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