你将学到什么
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.
课程概况
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.
课程大纲
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
预备知识
Experience in college-level mathematics or computer-science maybe helpful.
常见问题
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.
