Incompleteness

Kurt Godel 1906 — 1978

Godel spent his life working in the arcane field of mathematical logic and set theory. In 1931 he published his most famous work, the Incompleteness Theorem.

This states that within a formal mathematical system there are propositions that can be neither proved nor disproved on the basis of the axioms underpinning that system. That is, some statements are inherently ‘undecidable’. Therefore, one cannot be sure that the ‘basic axioms of arithmetic will not give rise to contradictions’.

This was very upsetting to many mathematicians (not least Bertrand Russell having just completed his Principia Mathematica supposedly proving the opposite) who believed that mathematics was entirely self-consistent and that everything they ‘proved’ was therefore true. (It also implies that a computer can never be programmed to answer all mathematical questions…)