On the independence of the axioms of definiteness (Axiome der Bestimmtheit)

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The question of the independence of the axioms of the theory of sets has been dealt with in a number of works, although not in a final manner. The writer will be concerned solely with the axiomatic system of Zermelo and Fraenkel, and only with that feature of the system whereby all the objects of the underlying domain are sets (so that there is no difference between objects in general and sets in particular).A special place among the axioms is occupied by a purely relational one, an axiom of definiteness, which establishes the character of equality within the system.Zermelo introduces equality intensionally—if two symbolsxandyrepresent the same object, we write = (x, y). According to this there is no necessity for axioms to assure the interchangeability of equal objects as arguments of the primitive relationϵ( , ). For under the intensional interpretation it is clear that:Similarly it is clear that there is no necessity for a separate axiom that will assure to equality the properties of an equivalence relation. Therefore Zermelo introduces only the followingaxiom of extensionality:(if one set is a subset of another set, and the second set is also a subset of the first, then the two sets are equal).