The adventure continued with the growing realization that all of mathematics can, in principle, be reduced to the theory of sets coupled with a few simple rules of logical deduction. This realization weathered some scary moments occasioned by Russell's paradox and Goedel's incompleteness theorems, and it is now almost universally accepted.
This course is designed to give students a good grounding in set theory that will give them a thorough understanding of these early events, as well as some insight into later phases of the great adventure (see below). We will mostly be following Judith Roitman's Introduction to Modern Set Theory but will supplement it with handouts from other sources, and with lectures on some interesting discoveries from these first phases of the adventure, including an example of a subset of the real line whose total length is too small to be any number greater than zero, yet too big to be zero.
Roitman's textbook seems to be just the right level for this course, and should be accessible to seniors who are eligible to take graduate courses. The book is out of print but the author (to whom the copyright reverted) has given permission to photocopy any or all of it for class use, so anyone taking this course will get a financial break. The book has lots of routine exercises which are designed to get the reader comfortable with the concepts. Homework, which counts for most of the course grade, will be taken mostly from Roitman's book. There is a copy on reserve in the math library on the third floor, under the counter behind which the librarians work.
Goedel began a new phase of the adventure when he showed in 1939 that the CH is consistent with the usual axioms of set theory. The usual way of saying this in popular accounts is that CH could be added to the the generally accepted axioms of set theory without leading to a contradiction, as long as the generally accepted axioms themselves are consistent. Then in 1963 Cohen began a revolution in set theory when he showed that (as it is often phrased) the denial of the CH can also be added to the generally accepted axioms without any fear of introducing contradictions. Since then, a multitude of other statements, not just in set theory but also in topology, algebra, and analysis, have been shown to be undecidable by all the generally accepted axioms. I have written a short expository paper with a lavishly annotated bibiligraphy, recently published electronically by Topology Atlas, that tells a little more about all this.
This course will give some idea of what all this means and how such things can be proven, but there will not be enough time to actually work through the proofs "from scratch." But we will see some applications of various undecidable axioms, including CH.