In 1900 at the start of the twentieth century, he gave an address to the worlds mathematicians. In it he posed twenty three still unsolved mathematical problems and predicted that this century would be spent in solving them. There are still three problems unsolved__.
But we are interested in his work on the Foundations of mathematics.
He was worried about the difficulties posed by Russell i.e. is mathematics true and he was also worried by the constuructionalists.