Lucas argued that a thinker can recognize the truth of its own Gödel sentence, while no consistent formal system can prove its own such sentence—this asymmetry is not incidental but definitional of rational agency.

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Consistent formal system(as used in logic and mathematics): A set of logical rules and symbols where you can't prove something is both true and false at the same time—there are no contradictions built into the system.
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 sentence(the main subject of the statement): A mathematical statement (named after logician Kurt Gödel) that is true but cannot be proven using the rules of a particular logical system.
Lucas(as a philosopher in debates about artificial intelligence and mathematics): John Lucas, a British philosopher who argued that human minds can do things that computers cannot, based on Gödel's mathematical discoveries.

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