The concept of infinity has intrigued and puzzled mathematicians for centuries. Near the end of the 19-th century, Cantor made the shocking discover that not all infinite sets have the same size. Shortly afterwards, Russell found a paradox in Frege's system of set theory and showed that mathematicians did not understand the idea of a "set" as well as they thought they did. Together, these discoveries helped launch the modern study of set theory and of infinite sets.

In this talk, we will discuss these ideas and examine related questions. How can we measure the size of a set? How can one infinite set be bigger than another infinite set? How big is the set of real numbers? This last question is one that is still debated today!