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Natural models and Ackermann-type set theories1

  • John Lake (a1)


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 xRα, we use Df(Rα, x) for the set of those elements of which are definable in 〈, ∈〉, 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.



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Most of the results in this paper were included in part 1 of the author's Ph.D. thesis [4], which was presented to the University of London. Dr. John Bell is thanked for a number of helpful conversations about the topics which we consider and the Science Research Council is thanked for its financial support.



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[1]Ackermann, W., Zur Axiomatik der Mengenlehre, Mathematische Annalen, vol. 131 (1956), pp. 336345.
[2]Grewe, R., Natural models of Ackermann's set theory, this Journal, vol. 34 (1969), pp. 481488.
[3]Lake, J., On an Ackermann-type set theory, this Journal, vol. 38 (1973), pp. 410412.
[4]Lake, J., Some topics in set theory, Ph.D. Thesis, University of London, 1973.
[5]Mostowski, A., Constructible sets with applications, North-Holland, Amsterdam, 1969.
[6]Novak, J., A paradoxical theorem, Fundamenta Mathematicae, vol. 37 (1950), pp. 7783.
[7]Reinhardt, W., Ackermann's set theory equals ZF, Annals of Mathematical Logic, vol. 2 (1970), pp. 189249.

Natural models and Ackermann-type set theories1

  • John Lake (a1)


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