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What we really wish to do is to take as the basis of our considerations the proposition: ‘Every ordinal is the type of the set of all ordinals that precede it’. But in order to avoid the vague notion ‘type’, we express it in the form: ‘Every ordinal is the set of the ordinals that precede it’. (von Neumann 1923, p. 347 of the English translation)

According to von Neumann's idea, 1 is just {0}, 2 is just {0, 1}, 3 is just {0, 1, 2} and so on. On this conception, the first transfinite ordinal ω is just {0, 1, 2, 3, …, n, …}, and generally it's clear that the immediate successor of any ordinal α is just α ∪ {α}. If we identify 0 with Φ, as Zermelo did, then we have available a reduction of the general notion of ordinal to pure set theory, where the canonical well-ordering on the von Neumann ordinals is just the subset relation, i.e., α < β just in case α ⊂ β, which von Neumann later shows is itself equivalent to saying α ∈ β. (See von Neumann 1928, p. 328 of the reprinting.) So again, inclusion orderings are fundamental.

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 Φ, {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

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