Axiomatic Set Theory in memoriam

In the last hundred-odd years, set theory has been studied mainly as axiomatized mathematical theory. This axiomatic approach to set theory was launched by Zermelo (1908b; see also Ebbinghaus 2007). Under Hilbert‘s influence, he presented in 1908 a set of axioms for set theory. His main objective can be said to have been to remove the uncertainties from the set-theoretical foundations of mathematics that the paradoxes had inflicted on mathematicians. Zermelo‘s other motive, a more specific one, was to buttress the axiom of choice that he had used in his well-ordering proofs (Zermelo 1904, 1908a). Zermelo‘s first axiom system was not fully formal, but one can find there essentially the same (or the corresponding) axioms that were later explicitly formulated and incorporated in the now widely used Zermelo-Frankel axiomatization of set theory (ZF; cf. Fraenkel 1919). They included prominently a suitably restricted form of the axiom of comprehension that in its completely unrestricted version could be formulated as the following schema