Zermelo s1932d

pp. 564-571

in: Ernst Zermelo, Set theory, miscellanea / Mengenlehre, varia, Berlin, Springer, 2010

Abstract

Set theory is concerned with those mathematically defined infinite totalities or domains which are called "sets" and among which the "finite" ones only occur as a special borderline case. Since an infinite totality can never be given or presented empirically, the definition of such a domain can always only proceed axiomatically through the specification of a system of conditions that this ideally posited infinite domain, which only exists as an idea in Plato's sense, is supposed to satisfy. Examples of such axiomatically defined infinite totalities or sets are the system of the natural numbers in the sense of Peano's postulates and the system of the reals in the sense of Hilbert's axioms.