date_range Starts on June 18, 2019
event_note End date September 10, 2019
list 12 sequences
assignment Level : Intermediate
chat_bubble_outline Language : English
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credit_card Free access
verified_user Fee-based Certificate
timer 60 hours in total

About the content

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 both the MITx Philosophy Award and 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.

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Experience in college-level mathematics or computer-science may be helpful.



Module 1: INFINITY

Week 1 Infinite Cardinalities

Week 2 The Higher Infinite

Week 3 Omega-Sequence Paradoxes



Week 4 Time Travel

Week 5 Newcomb’s Problem

Week 6 Probability

Week 7 Non-Measurable Sets

Week 8 The Banach-Tarski Theorem



Week 9 Computability

Week 10 Gödel’s Theorem


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

Cosmo Grant
Course Coordinator
Massachusetts Institute of Technology


Content designer

MIT is a world-class educational institution where teaching and research — with relevance to the practical world as a guiding principle — continue to be its primary purpose.

MIT is independent, coeducational, and privately endowed. Its five schools and one college encompass numerous academic departments, divisions and degree-granting programs, as well as interdisciplinary centers, laboratories and programs whose work cuts across traditional departmental boundaries.



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