Set theory (Math 320)
Spring semester, 2008/9
The main text for this course is the lectures. There is also a supplementary text, called Sets, Classes, and Families: Notes on set theory, obtainable in two ways:
This text is a thorough revision of the text used last time I taught this course. The preface mentions some useful books; and there are many others.
Again though, the main text is the lectures. In the lectures, I shall not cover everything in the printed notes; this depends in part on the interest of the class. Moreover, the lectures need not have the same order as the printed notes. Finally, I cannot guarantee that everything that will be in the lectures is covered in the notes (although this was my intention in writing the notes).
General remarks about set theory
I recall here the catalogue-description of the course:
Language and axioms of set theory. Ordered pairs, relations and functions. Order relation and well ordered sets. Ordinal numbers, transfinite induction, arithmetic of ordinal numbers. Cardinality and arithmetic of cardinal numbers. Axiom of choice, generalized continuum hypothesis.
Set-theory is useful to mathematics in the following way. We establish certain properties of sets, usually by means of axioms, such as the so-called Zermelo–Fraenkel system of axioms. Then we can define various standard mathematical structures (such as the ordered field ℝ of real numbers), and we can prove the various properties of these structures, without having to introduce new assumptions. In this way, set-theory provides a uniform foundation for mathematics.
Set-theory is also worth pursuing for its own sake. The development of set-theory involves the creation of certain new structures (such as the class ON of ordinal numbers, or the class L of constructible sets) that are interesting in themselves. Set-theory gives us certain useful techniques (such as various forms of proof by induction and definition by recursion).
Number-theory contains theorems (such as the so-called Fermat's Last Theorem) that are easy to understand, but almost impossibly difficult to prove. Some theorems of set-theory are harder to understand (and also difficult to prove), but are mind-blowing in their connotations. For example, ℝ is strictly greater in cardinality than the set ω of natural numbers; in a word, ℝ is uncountable; however, the Zermelo–Fraenkel axioms do not determine whether ℝ has the least uncountable cardinality.