Natural models and Ackermann-type set theories

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 ⊆ R α, 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.