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What Gödel’s incompleteness theorems say
Take any set of rules for proving things in math, as long as the rules are clear enough that a computer could check a proof, and strong enough to do basic arithmetic. Gödel showed two things about such a system.
First, it cannot prove everything that is true. There are always true statements about numbers that the rules can never reach. Second, it cannot prove that it is itself consistent, meaning that it will never contradict itself.