Lesson 8

In 1930, the twenty-four-year-old mathematician Kurt Gödel shocked the world by proving that any attempt to axiomatize all of mathematics was guaranteed to fail. In short, if a set of axioms is sound (i.e. if every statement that can be logically derived from the axioms is true) and strong enough to prove some basic facts, then there must be statements that can neither be proven nor disproven under the axioms.

Theorem (Gödel, 1931). In any consistent axiom system within which a certain amount of arithmetic can be carried out, there are statements which can neither be proved nor disproved in this system.