 The choice of Zermelo set theory (leaving aside whether one goes on to ZFC) rules out the use of equivalence classes of equinumerous sets as cardinals (and so the use of the Frege natural numbers) or the use of equivalence classes of well-orderings as ordinals. There is no difficulty with the use of the Dedekind cut formulation of the reals (once the rationals have been introduced). Instead of the equivalence class formulations of cardinal and ordinal numbers, the von Neumann ordinals are used: a von Neumann ordinal is a transitive set (all of its elements are among its subsets) which is well-ordered by membership. The order type of a well-ordering is the von Neumann ordinal of the same length (the axiom of Replacement is needed to prove that every set well-ordering has an order type; this can fail to be true in Zermelo set theory, where the von Neumann ordinal \(\omega + \omega\) cannot be proven to exist but there are certainly well-orderings of this and longer types). The cardinal number \(|A|\) is defined as the smallest order type of a well-ordering of \(A\) (this requires Choice to work; without choice, we can use Foundation to define the cardinal of a set \(A\) as the set of all sets equinumerous with \(A\) and belonging to the first \(V_{\alpha}\) containing sets equinumerous with \(A)\). This is one respect in which Cantor’s ideas do not agree with the modern conception; he appears to have thought that he could define at least cardinal numbers as equivalence classes (or at least that is one way to interpret what he says), although such equivalence classes would of course be Absolutely Infinite.