Gödel's first incompleteness theorem

The first theorems of Gödel’s incompleteness theorems states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.

See also

• Gödel’s second incompleteness theorem