### Agustín Rayo

Professor of Philosophy and Associate Dean of SHASS Massachusetts Institute of Technology

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## About this course

## What you'll learn

## Syllabus

## Meet your instructors

### Agustín Rayo

### David Balcarras

### Cosmo
Grant

24.118x
#
Paradox and Infinity

This course is currently available on MITx Online with Certificate enrollment open.

In *Paradox and
Infinity*, you will be introduced to highlights from the intersection of philosophy
and mathematics.

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.

Learners who display exceptional performance in the class are eligible to win the MITx
Philosophy Award. High School students are eligible for that award and, in addition, the
MITx High School Philosophy award. **Please see the FAQ section below for additional
information.**

Note: learners who do well in Paradox will have typically taken at least a couple of college-level classes in mathematics or computer science. On the other hand, Paradox does not presuppose familiarity with any particular branch of mathematics or computer science. You just need to feel comfortable in a mathematical setting.

- 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.

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

Professor of Philosophy and Associate Dean of SHASS Massachusetts Institute of Technology

Digital Learning Lab Fellow Massachusetts Institute of Technolog

Digital Learning Lab Fellow Massachusetts Institute of Technology