Meeting times: M,W from 3:00 to 4:10pm
Text: Kunen, Set Theory (Studies in Logic edition)
Notes: View notes (will be updated as the course evolves)
Homework: View homework
Office hours: TBA
This course is about sets of real numbers, and more specifically about mesauring the size of sets of real numbers. But the word ``size’’ means different things to different mathematicians: three important examples are cardinality, measure, and category. Cardinality asks whether a set is countable or uncountable; measure asks whether a set has zero length or nonzero length, and category asks whether a set is meager or nonmeager. We will ask: how do these three kinds of size compare with one another?
Our investigation of mesaure and category will lead us to a number of independence results, that is, statements that cannot be decided using the usual axioms of set theory. The most central example of such a statement is the continuum hypothesis, which asserts that the set of all real numbers has cardinality equal to the first uncountable cardinal number. In order to prove this statement is indeed independent of set theory, we will study a technique called set-theoretic forcing.
Once we have developed the machinery of forcing, we will return to the notions of measure and category. We will find that there are numerous cardinal numbers surrounding the zero length sets and the meager sets whose values, like the size of the set of all real numbers, are also independent of set theory.
Homework problems will be assigned weekly and collected the following week (Tuesday night). To receive a C, you must achieve a base level of success on every single problem. To receive a B, most of solutions should be correct and complete. To receive an A, at least 3/4 of your solutions should be correct and complete. You may correct and resubmit homework problems within a couple weeks of receiving them back.
Attendance and participation are encouraged. Additionally you may be asked to give short presentations about homework solutions or other class material.