On the consistency and independence of some set-theoretical axioms.
Alexander Abian, Samuel E. Lamacchia · Notre Dame Journal of Formal Logic · 1978
In this paper by means of simple models it is shown that the five set-theoretical axioms of Extensionalit y, Replacement, Power-Set, SumSet, and Choice are consistent and that each of the axioms of Extensionality, Replacement, and Power-Set is independent from the remaining four axioms. Although the above results are known and can be found in part in [l], it is believed that this paper has some expository merits. The abovementioned axioms are five of the six axioms of the ZermeloFraenkel Theory of Sets, the sixth being Axiom of Infinity. We accept that every element of a set is a set and (without borrowing from Logic) we define two sets u and υ to be equal if and only if they possess the same elements, in which case we write u = v. With this in mind, the six axioms of the Zermelo-Fraenkel Theory of Sets can be stated as follows [2]: