Which mathematician's incompleteness theorems shook logic in 1931?

Kurt Gödel, at 25, who proved that any consistent formal system rich enough to describe arithmetic contains true statements it cannot prove - and cannot prove its own consistency. It arrived while Hilbert's programme was attempting to put all of mathematics on a complete, provably consistent foundation, and it ended that project permanently.

The method is the beautiful part. Gödel encoded statements about arithmetic as numbers, so a formal system could talk about itself, and then constructed a statement that says in effect this statement is not provable. If it is provable the system is inconsistent; if it is not, it is true and unprovable. It is the liar paradox made rigorous.

He became close friends with Einstein at Princeton, who said he went to the institute mainly for the privilege of walking home with Gödel. He was also severely paranoid, convinced he would be poisoned, and would eat only food his wife prepared. When she was hospitalised for six months he stopped eating and died weighing 29 kilograms - a man who had proved the limits of formal certainty, destroyed by an absolute refusal to trust.