Reckoning with infinity is one of the major accomplishments of mathematics. Start with basic counting and work your way up to the many types of infinity, culminating in the profundity of Cantor's theorem. Explore the stunning beauty of fractals, where it's turtles all the way down, and tesselations in hyperbolic space, where the infinitely large is bounded by a simple circle.
Overview
Syllabus
- Introduction: Tangle with the mathematical idea of the infinite.
- What's Infinite?: What do we really mean when we use the word "infinity?"
- How to Count to Infinity: Reconsider basic counting to handle infinite sets of things.
- Visualizing Infinity: While we can't draw what infinity really looks like, we can make a good attempt!
- Sizes of Infinity: Infinity has a "size". How can we compare it with other infinities?
- Cardinality: Establish one of the basic concepts of set theory.
- Countably Infinite: Some infinite sets are countable in the same way.
- Cantor's Quest: Are all infinite sets countable?
- Hilbert's Infinite Hotel: Help Hilbert use his hotel that has infinitely many rooms to host infinitely many sleepy guests.
- Subsets: Tangle with sets of things inside of sets of things.
- Cantor's Theorem: Cantor blows open the universe of infinity in one of the deepest theorems of mathematics.
- Visual Infinities: Represent the infinite in visuals and gain new insights.
- Infinite Areas: Quantify the infinite using geometry.
- Infinite Sums: Use infinite area problems to derive the basic ideas behind infinite sums.
- The Koch Snowflake: Grow this course's first fractal.
- More Fractals: Making copies of copies of copies of copies of copies of copies...
