How Big Is Infinity? And Other Math Questions
This course is expected to run but has not yet been scheduled.
Have you ever wondered how many numbers there really are or how big infinity is? Many mathematicians have asked these questions, and the answers can be fascinating . . . and sometimes quite strange. This course will be a journey through the mathematical concepts that have helped answer such questions, and along the way, you'll get a taste of what pure mathematics is all about.
We will cover the tools necessary to tackle such difficult questions, starting with an introduction to some mathematical ideas including functions, set theory and number sets, cardinality, and different notions of infinity, as well as some unexpected results and paradoxes that come from them. An emphasis will be placed on mathematical logic and proof, and these concepts will be illuminated through a variety of problems and thought exercises. We will also talk a little about the history of the idea of infinity and the mathematicians behind it. Many of the mathematical topics will be essential for anyone wanting to be a math major in college, particularly in the fields of abstract algebra and analysis. More generally, they are useful for anyone who wants to study a subject requiring critical thought or thinking outside the box.
The ultimate goal of this course is to enable students to understand and engage in fundamental ideas in mathematical and logical thought and proof writing. At the end of the course, students will be familiar with the notion of a proof in mathematics and logic, and have a deeper appreciation of mathematics, not as a mere abstraction, but as a field which can shed light on seemingly familiar, yet ethereal ideas such as infinity.
As for prerequisite knowledge, algebra will be assumed.