Our results concern the natural models of Ackermann-type set theories, but they can also be viewed as results about the definability of ordinals in certain sets.
Ackermann's set theory A was introduced in [1] and it is now formulated in the first order predicate calculus with identity, using ∈ for membership and an individual constant V for the class of all sets. We use the letters ϕ, χ, θ, and χ to stand for formulae which do not contain V and capital Greek letters to stand for any formulae. Then, the axioms of A* are the universal closures of
where all the free variables are shown in A4 and z does not occur in the Θ of A2. A is the theory A* − A5.
Most of our notation is standard (for instance, α, β, γ, δ, κ, λ, ξ are variables ranging over ordinals) and, in general, we follow the notation of [7]. When x ⊆ Rα, we use Df(Rα, x) for the set of those elements of Rα which are definable in 〈Rα, ∈〉, using a first order ∈-formula and parameters from x.
We refer the reader to [7] for an outline of the results which are known about A, but we shall summarise those facts which are frequently used in this paper.