Kurt Gödel shook the mathematical world in 1931 with his incompleteness theorems. These revolutionary results demonstrated fundamental limitations of formal mathematical systems.

The First Incompleteness Theorem states that for any consistent formal system F within which basic arithmetic can be carried out, there are statements that can be formulated in F that can neither be proved nor disproved within F.

This means that mathematics contains true statements that cannot be proved within the system itself - a profound and somewhat unsettling result that changed our understanding of mathematical truth and formal systems forever.

The implications of this theorem continue to influence fields ranging from computer science to philosophy of mathematics.