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  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 . 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  for an outline of the results which are known about A, but we shall summarise those facts which are frequently used in this paper.