In 1931, the 25-year-old Austrian logician Kurt Gödel published a paper called 'On Formally Undecidable Propositions of Principia Mathematica and Related Systems.' It was short and densely technical, and it broke a more-than-2,000-year-old assumption about mathematics.

The assumption was that mathematical truth was, at bottom, the same as mathematical proof. Every true statement, the thinking went, could in principle be derived from a sufficiently good set of starting axioms. In the early 20th century, the German mathematician David Hilbert had laid out a formal program to prove this — to put all of mathematics on a complete and consistent axiomatic foundation, with every truth eventually reachable by mechanical deduction.

Gödel showed that the program could not succeed. His First Incompleteness Theorem states, roughly, that any consistent formal logical system powerful enough to describe ordinary arithmetic must contain true statements that cannot be proven within that system.

His method was unusual and elegant. He developed a technique now called Gödel numbering, which encoded statements about a logical system as ordinary numbers inside the same system. Using this, he constructed a self-referential statement, often paraphrased as 'this statement cannot be proven within this system.' If the system proved the statement, it contradicted itself. If it could not prove the statement, the statement was true — but unproven. Either way, the system had a permanent hole.

A few months later, he proved a Second Incompleteness Theorem: no sufficiently powerful consistent system can prove its own consistency from inside itself.

Hilbert's program, as originally formulated, was finished. The implications spread. Alan Turing's 1936 paper on computability, which laid the theoretical foundation for the modern computer, used a closely related construction to prove the unsolvability of the halting problem. Generations of philosophers have argued about what Gödel's result implies for the nature of human reason and the limits of artificial intelligence.

Almost a century later, the hole he revealed is still there. There are mathematical truths that no formal proof system can reach.