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Concerning the axioms of Ackermann’s set theory. (English) Zbl 0564.03039
In the paper there are considered some foundational aspects of Ackermann’s set theory (A) [cf. A. Fraenkel, Y. Bar-Hillel and A. Levy, Foundations of set theory, 2nd ed. (1973; Zbl 0248.02071)]. A and its subtheory, resulting from dropping the axioms of extensionality and heredity, are proved to be deductively equivalent. There are discussed some interpretative difficulties concerning Ackermann’s axioms and their intended meaning. Some alternative axiom systems are proposed, and their relation to standard set theories (ZF and Z) is examined.
MSC:
03E70 Nonclassical and second-order set theories
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