Set Theory and Foundations of Mathematics

Set theory was created almost single-handedly by Cantor in the latter part of the 19th century. Cantor's motivation came from analysis, in particular the problem of representability of functions by Fourier series. Set Theory maintained its close connection with analysis but in time became an autonomous subject. Paradoxes in set theory appeared in the early 20th century, leading in due course to the subject's axiomatization. Set theory began to serve as the foundation and language for much of mathematics. (For example, the fundamental concepts of relation and function are rigorously defined in terms of sets.) But it has also evolved into a flourishing discipline in its own right. Moreover, some of the groundbreaking results it has established in this century (e.g., Godel's Incompleteness Theorems, Cohen's Independence results) have had far-reaching philosophical implications. The main points in the historical account of set theory sketched above will serve as guideposts around which to structure the course. Thus, the origins of set theory will be touched on, followed by the "naive" development of some of its main notions, including cardinal and ordinal arithmetic, well ordering, and the axiom of choice. Several paradoxes in set theory will be noted, followed by an outline of the axiomatization of set theory. It will then be shown how this leads to a rigorous development of some of the basic notions of mathematics such as function and the natural numbers. Issues in the philosophy of mathematics, including formalism, intuitionism, and logicism, the problems of consistency, and Godel's theorems will also be discussed. The final grade will be based on assignments and tests (60%) and a final exam (40%). The text for the course has not yet been determined.