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Paradox and Infinity

This is a class about awe-inspiring issues at the intersection between philosophy and mathematics.

This course is archived
Estimated 12 weeks
5–6 hours per week
Instructor-paced
Instructor-led on a course schedule
Free
Optional upgrade available

About this course

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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.

At a glance

  • Institution: MITx
  • Subject: Philosophy & Ethics
  • Level: Intermediate
  • Prerequisites:

    Experience in college-level mathematics or computer-science maybe helpful.

  • Language: English
  • Video Transcript: English

What you'll learn

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  • 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

About the instructors

Frequently Asked Questions

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Verified learners will be eligible for the MITx Philosophy Award and the MITx High School Philosophy Award. The awards will recognize outstanding work and will be certified by the MIT Philosophy department.

Please visit the MIT Philosopy Department website for additional information on the Award and eligibility on the Philosophy Prize, as well as a list of winners from past years: http://web.mit.edu/philosophy/mitx.html.

Who can take this course?

Unfortunately, learners residing in one or more of the following countries or regions will not be able to register for this course: Iran, Cuba and the Crimea region of Ukraine. While edX has sought licenses from the U.S. Office of Foreign Assets Control (OFAC) to offer our courses to learners in these countries and regions, the licenses we have received are not broad enough to allow us to offer this course in all locations. edX truly regrets that U.S. sanctions prevent us from offering all of our courses to everyone, no matter where they live.

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