2 Set Theory as a Foundation

Author Webpage The view of set theory as a foundation for mathematics emerged early in the thinking of the originators of the theory and is now a pillar of contemporary orthodoxy. As such, it is enshrined in the opening pages of most recent textbooks; to take a few illustrative examples: All branches of mathematics are developed, consciously or unconsciously, in set theory. (Levy (1979), 3) Set theory is the foundation of mathematics. All mathematical concepts are defined in terms of the primitive notions of set and membership . . . From [the] axioms, all known mathematics may be derived. (Kunen (1980), xi) [M]athematical objects (such as numbers and differentiable functions) can be defined to be certain sets. And the theorems of mathematics (such as the fundamental theorem of calculus) then can be viewed as statements about sets. Furthermore, these theorems will be provable from our axioms. Hence, our axioms provide a sufficient collection of assumptions for the development of the whole of mathematics—a remarkable fact. (Enderton (1977), 10–11)