We prove the following theorem: If S is a consistent set of axioms for set theory, among whose consequences are the axioms of pairing and extensionality, and if S′ is the extension of S to all classes of sets of S, then S′ is consistent and remains so upon addition of an axiom expressing that the universal class has a well-ordering wherein every non-initial element has an immediate predecessor.
The basic idea will be use of the denumerable model of S guaranteed by the Skolem-Löwenheim theorem to obtain (i) a model of S′ in the domain of classes of natural numbers, and (ii) a correspondence between representatives (in the model of S′) of sets of S′ and natural numbers. One can then select a class of natural numbers in the model of S′ which will well-order (in the sense of the model of S′) the representatives of sets of S′ according to the numbers corresponding to them.
Specifically, let the formulas of S involve only one kind of variable and the sole predicate ‘ϵ’. The formulas of S′ can be thought of as identical with those of S; but the axioms of S′ will consist of (i) those of S with quantifiers restricted to sets by clauses ‘(∃z)[y ϵ z]’, (ii) those provided by the schema ‘(∃x) ▪ y ϵ x ≡y [(∃z)[y ϵ z] ▪ A]’, where ‘A’ is any formula of S′ in which ‘x is not free, and (iii) the axiom of extensionality.
