Paul Cohen

Paul Cohen (1934–2007) was an American mathematician with a reputation for fearlessly attacking the hardest problems he could find — and winning. He is the only person ever to receive the Fields Medal, maths' top honour, for work in mathematical logic. His weapon was a brand-new technique he invented practically from scratch.

The enduring contribution

Gödel had shown you couldn't disprove the continuum hypothesis — the question of whether there's any size of infinity strictly between the whole numbers and the real numbers. In 1963 Cohen delivered the other half: you can't prove it either. Together their results mean the continuum hypothesis is independent of the standard rules of mathematics — a genuinely undecidable question. To do it he invented "forcing," a method so powerful it's now a standard tool across all of set theory.

What makes the achievement even sweeter is that Cohen wasn't originally a set theorist at all — he was an analyst who essentially decided to go and crack the most famous open problem in logic, and taught himself the field to do it. He reportedly first suspected the continuum hypothesis might be provable, and only in wrestling with it discovered the truth was stranger: the question has no answer. He took his result to Gödel himself for the final blessing before publishing.