Gödel's Theorems (C/N)

Gödel’s incompleteness theorems are two theorems of mathematical logic which establish inherent limitations in all but the most trivial axiomatic systems capable of arithmetic. Proven by Kurt Gödel in 1931, they are both important to mathematical logic and the philosophy of mathematics. The two results are widely interpreted as showing Hilbert’s program to find a complete and consistent set of axioms for all mathematics is impossible. The first theorem states that no consistent system of axioms can be listed by an “effective procedure,” and the second (as an extension of the first) shows that such a system cannot demonstrate its own consistency.