The problem was fully solved by von Neumann in his work on axiomatic set theory from the early 1920s. Cantor's fundamental theorems about ordinal numbers, showing that the ordinals are the representatives of well-ordered sets, are the theorem that every well-ordered set is order-isomorphic to an initial segment of the ordinals, and that every ordinal is itself the order-type of the set of ordinals which precede it. These results prove crucial in the von Neumann treatment. Von Neumann's basic idea was explained by him as follows: