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On an Ackermann-type set theory

  • John Lake (a1)


Ackermann's set theory A* is usually formulated in the first order predicate calculus with identity, ∈ for membership and V, an individual constant, for the class of all sets. We use small Greek letters to represent formulae which do not contain V and large Greek letters to represent any formulae. The axioms of A* are the universal closures of

where all free variables are shown in A4 and z does not occur in the Θ of A2.

A+ is a generalisation of A* which Reinhardt introduced in [3] as an attempt to provide an elaboration of Ackermann's idea of “sharply delimited” collections. The language of A+ is that of A*'s augmented by a new constant V′, and its axioms are A1–A3, A5, VV′ and the universal closure of

where all free variables are shown.

Using a schema of indescribability, Reinhardt states in [3] that if ZF + ‘there exists a measurable cardinal’ is consistent then so is A+, and using [4] this result can be improved to a weaker large cardinal axiom. It seemed plausible that A+ was stronger than ZF, but our main result, which is contained in Theorem 5, shows that if ZF is consistent then so is A+, giving an improvement on the above results.



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[1]Church, A., Set theory with a universal set, Proceedings of the Tarski Symposium, American Mathematical Society, 1973, pp. 291302.
[2]Levy, A., On Ackermann's set theory, this Journal, vol. 24 (1959), pp. 154165.
[3]Reinhardt, W., Ackermann's set theory equals ZF, Annals of Mathematical Logic, vol. 32 (1970), pp. 189249.
[4]Reinhardt, W. and Silver, J., On some Problems of Erdös and Hajnal, Notices of the American Mathematical Society, vol. 12 (1965), p. 723. Abstract 65Y-445.

On an Ackermann-type set theory

  • John Lake (a1)


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