Gödel's Incompleteness Theorems

tags:: #a/concept Philosophy on/mathematics

The first incompleteness theorem 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.

The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.

A set of axioms is (simply) consistent if there is no statement such that both the statement and its negation are provable from the axioms, and inconsistent otherwise. That is to say, a consistent axiomatic system is one that is free from contradiction.