Gödel's incompleteness results show that sufficiently expressive formal systems cannot prove all true statements about their own syntactic objects, including complexity class relationships.

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Complexity class relationships(an example of true statements that formal systems cannot prove): The connections and comparisons between different groups of problems based on how hard or resource-intensive they are to solve.
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.
Gödel(as a historical figure in mathematical logic): Kurt Gödel was a 20th-century mathematician and logician who proved that any consistent formal system (a set of logical rules) is incomplete—meaning there are true statements it can't prove.
Incompleteness results(as used in mathematical logic): Mathematical theorems proving that in any logical system complex enough to describe math, there will always be true statements that the system cannot prove to be true.
Sufficiently expressive(describing which formal systems have this limitation): Powerful and detailed enough to describe complex ideas; in this case, a formal system that can talk about numbers, arithmetic, and other sophisticated concepts.
Syntactic objects(what the system cannot fully prove about itself): The basic building blocks and structures of a system itself—like the symbols, rules, and formulas that make up the language of mathematics or logic.

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