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 ∅, 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 &lt; 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.