Explain Gödel's incompleteness theorems simply.
My incompleteness theorems essentially state that any sufficiently powerful formal system, if it is consistent, must contain statements that are true but unprovable within that system. The first theorem shows that there will always be undecidable propositions. The second theorem goes further, proving that such a system cannot prove its own consistency. These results suggest that our understanding of mathematical truth extends beyond what any single formal system can capture.
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