Cantor's Diagonal Argument
There are more numbers between 0 and 1 than there are whole numbers in all of eternity. Both lists are infinite.
189116 min read
German mathematics
David Hilbert
1862 – 1943
The dominant mathematician of his generation and the centre of gravity of Göttingen until the Nazi purges emptied it. His 1900 list of twenty-three problems set much of the century's agenda, and his defence of Cantor's transfinite mathematics — "no one shall expel us from the paradise that Cantor has created" — supplied both the formalist programme and the hotel that carries his name.
3 entries
Hilbert's Grand Hotel
A hotel with infinitely many rooms is completely full. At two in the morning, one more guest arrives. The manager finds them a room.
1924 (lectures), 1947 (popularised)13 min read
Gödel's Incompleteness Theorems
In any system of arithmetic strong enough to be useful, there are true statements it can never prove. Including this one.
193125 min read