Gödel's Incompleteness Theorem

The Incompleteness Theorem states a limitation on what we can know.

Gödel's incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating the inherent limitations of all but the most trivial formal systems for arithmetic.

During Gödel's lifetime the German mathemetician Hilbert was pursuing a complete and consistent set of axioms for all of mathematics. Gödel's work states that such a set is impossible.

Gödel's first theorem:
For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true but not provable in the theory can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

According to Gödel, any set of axioms and rules must contain true statements that cannot be proven within that set. It may be possible to augment the set with new axioms and/or rules, but then you obtain a new set which is larger but still incomplete having statements that are true but unprovable within the system. This means that any system of axioms and rules will contain statements that cannot be proven or disproven based solely on the rules and axioms of that system. This, by extension, means that you cannot be certain that the set of rules and axioms are consistent, meaning they will never lead to contradictory statements according to the rules of the system.

Gödel's second theorem:
For any formal recursively enumerable (i.e. effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.