Godel's Incompleteness Theorems

Godel's incompleteness theorems are two results in mathematical logic, proved by Austrian logician Kurt Godel in 1931, establishing fundamental limits on what any sufficiently powerful, consistent formal axiomatic system can prove. The first theorem shows that such a system will always contain true statements about arithmetic that cannot be proved within the system itself, while the second shows that the system cannot prove its own consistency using only its own axioms. The theorems ended a decades-long program, associated especially with David Hilbert, that had sought a complete and self-verifying foundation for all of mathematics, and they remain among the most significant results in twentieth century mathematical logic.