Constructing Cardinals from Below

If the initial segment Σ of Ω is a set, then it has a least strict upper bound S(Σ) ∈ Ω. Thus, for numbers α = S(Σ) and β = S(Σ′), α < β iff α ∈ Σ′; α = β iff Σ = Σ′; S(∅) is the least number 0 (although Cantor himself took the least number to be 1); if Σ has a greatest element γ, then α is its successor γ + 1; and if Σ is non-null and has no greatest element, then α is the least upper bound of Σ. The problem with the definition, of course, is in determining what it means for an initial segment to be a set. Obviously, not all of them are: for the totality Ω of all numbers is an initial segment, but to admit it as a set would yield S(Ω) < S(Ω), contradicting the assumption that Ω is well-ordered by <. Cantor himself understood this already in 1883. In his earlier writings, e.g. [1882], he had essentially defined a set ‘in some conceptual sphere’ such as arithmetic or geometry, to be the extension of a well-defined property. But in these cases, he was considering sets of objects of some type A, where being an object of type A is itself is not defined in terms of the notion of a set of objects of type A. But with his definition of the transfinite numbers, an entirely novel situation arises: the definition of Ω depends on the notion of a subset of Ω. Accordingly, he abandoned his earlier definition of set in [1883] and, in his later writings, he distinguished between those initial segments which are sets and those which are not in terms of his concept of ‘consistent multiplicity’; but that is just naming the problem, not solving it.