Kurt Gödel
Can mathematics be complete in itself?
That time, that place
Hilbert had a program: rebuild all of mathematics on a handful of axioms with no gaps, and prove the system free of contradiction. And he believed that any true statement must eventually be provable.
We must know, he said, and we shall know.
Why this question
What Gödel did was unexpected: he devised a way to assign each statement its own number.
Then talk about statements becomes talk about numbers. Mathematics acquires the ability to speak about itself.
What was found
With this tool he constructed a statement meaning: this statement cannot be proved within this system.
Work it through. If it can be proved, then a statement asserting its own unprovability has been proved, and the system contains a contradiction. If the system is free of contradiction, the statement cannot be proved — and then what it says is true.
There exist statements that are true and unprovable.
The second theorem went further: no such system can prove its own consistency from within.
Hilbert's program was over.
Whatever is true must eventually be provable
Numbering statements so that mathematics could speak about itself
The technique of self-reference carried on to Turing five years later. In his final years Gödel became convinced he was being poisoned, refused food, and died