Gödel's incompleteness theorems
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Gödel's incompleteness theorems demonstrate that, in mathematics, it is impossible to prove everything.
More specifically, the first incompleteness theorem states that, in any consistent formulation of number theory but the most trivial ones, there are valid boolean statements that cannot be meaningfully assigned a value of "true" or "false." The second incompleteness theorem states that number theory cannot be used to prove its own consistency.
Godel demonstrated this by encoding the liar's paradox into number theory itself, creating a well-formed mathematical statement that referred to itself as a false statement.
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