Gödel's incompleteness theorems
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This article is only a brief description of the subject, and is not intended to give a full explanation.
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This article is only a brief description of the subject, and is not intended to give a full explanation.
Check out the "see also" or "references" sections, or Wikipedia's article for more detail.
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 unprovable statements. 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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