Von Neumann gives a general definition of his ordinals, namely that a set α is an ordinal number if and only if it is a set ordered by inclusion, the inclusion ordering is a well-ordering, and each element ξ in α equals the set of elements in the initial segment of the ordering determined by ξ. This connects directly with Kuratowski's work in the following way. Suppose M is a well-ordered set which is then mirrored by an inclusion chain M in the power set of M. Then the first few elements of the inclusion chain will be the sets ∅, {a}, {a, b}, {a, b, c}, …, where a, b, c, … are the first, second, third …elements in the well-ordering of M. The von Neumann ordinal corresponding to M will also be an inclusion ordering whose first elements will be