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Gödel's Incompleteness Theorems

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📖Definition

Gödel's incompleteness theorems show that in any sufficiently powerful mathematical system, there exist true statements that cannot be proved or disproved.

📐Formulas

Con(F) → ∃ G (F \nvdash G ∧ F \nvdash ¬ G)

First incompleteness theorem (informal)

Con(F) → F \nvdash Con(F)

Second incompleteness theorem

✏️Examples

예제 1

Explain the idea of Gödel sentence.

📜History

Discovered by: Kurt Gödel (1931)

Response to Hilbert's program, showing complete formalization of mathematics is impossible.

⚡Applications

Foundations

Limits of formal systems

Computer Science

Halting problem, computability

Philosophy

Truth vs. provability

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