In 1900 in Paris, David Hilbert listed mathematics' 23 great problems. The core concern was this: can mathematics be built upon a system in which, starting from sound axioms, we can prove every true proposition in a finite number of mechanical steps? Russell and Whitehead's "Principia Mathematica" (1910–1913) was the concrete attempt at this programme: across a thousand pages of symbol manipulation they tried to reduce arithmetic to logic. In the 1920s Hilbert sharpened the programme — mathematics had to be consistent, complete, and decidable.

At 25, having just finished his doctorate at the University of Vienna, Kurt Gödel published a 25-page paper in 1931 in "Monatshefte für Mathematik und Physik": "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I." It contained two theorems. The first: in every sufficiently strong, consistent formal system such as Peano arithmetic, there are propositions that can neither be proved nor refuted within the system's own tools, yet are nevertheless true. The second: no such system can prove its own consistency from within. His method was elegant and ruthless: he assigned an integer (a "Gödel number") to every mathematical expression so that the system could speak about its own sentences; then he constructed a sentence that says "this sentence is unprovable." If the sentence is false it is provable; but the moment it is proved, the system's consistency collapses. So it must be true — but unprovable.

The consequence was seismic. The Hilbert programme, in that form, was impossible; mathematics could never be sealed within a single formal system. But Gödel was not a destroyer, he was a boundary-drawer: he does not say mathematics is wrong or untrustworthy; he shows that no formal system built on finitely many rules can capture all of mathematical truth. Intuition, creativity and the acceptance of new axioms — these steps cannot be mechanised. John von Neumann would later say "this work stands unique in the history of mathematics and logic."

Gödel's technique — recursive enumeration, self-reference, formal reasoning about formal systems — turned, five years later in Alan Turing's hands, into the concept of "computability." The Turing machine, used to show the "decision problem" was unsolvable, became the theoretical foundation of the modern computer. The incompleteness theorems thus became the birth certificate not only of the philosophy of mathematics but of computer science, artificial intelligence debates and the philosophy of mind: from Penrose's claim that "the human mind is not an algorithm" to modern proof assistants, the discussion is still open.