The class is divided into three modules:
- Infinity: Learn about how some infinities are bigger than others, and explore the mind-boggling hierarchy of bigger and bigger infinities.
- Time Travel and Free Will: Learn about whether time travel is logically possible, and whether it is compatible with free will.
- Computability and Gödel’s Theorem: Learn about how some mathematical functions are so complex, that no computer could possibly compute them. Use this result to prove Gödel’s famous Incompleteness Theorem.
Paradox and Infinity is a math-heavy class, which presupposes that you feel comfortable with college-level mathematics and that you are familiar with mathematical proofs.
What you'll learn
- You will learn how to prove a number of beautiful theorems, including Cantor's Theorem, the Banach-Tarski Theorem, and Gödel's Theorem.
- You will acquire the ability to think rigorously about paradoxes and other open-ended problems.
- You will learn about phenomena at the boundaries of our theorizing, where our standard mathematical tools are not always effective.
Syllabus
Module 1: INFINITY
Week 1 Infinite Cardinalities
Week 2 The Higher Infinite
Week 3 Omega-Sequence Paradoxes
Module 2: DECISIONS, PROBABILITIES AND MEASURES
Week 4 Time Travel
Week 5 Newcomb’s Problem
Week 6 Probability
Week 7 Non-Measurable Sets
Week 8 The Banach-Tarski Theorem
Module 3: COMPUTABILITY AND GÖDEL’S THEOREM
Week 9 Computability
Week 10 Gödel’s Theorem