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Gödel's Incompleteness Theorems
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📖Definition
In any consistent formal system containing arithmetic, there exist true but unprovable statements. Also, the system cannot prove its own consistency.
📐Formulas
1st: There exists a true but unprovable statement
Incompleteness
2nd: System cannot prove its own consistency
Self-reference limitation
✏️Examples
예제 1
Why is Gödel sentence G: 'This statement is unprovable' true?
📜History
Discovered by: Kurt Gödel (1931)
Response to Hilbert's program; revealed fundamental limits of mathematics.
⚡Applications
Logic
Limits of proof theory
Computer Science
Undecidability, halting problem
Philosophy
Debates on nature of mathematics
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