Which problems can be solved by systematic algorithms, and which cannot? One of the marvelous qualities of mathematics is that it can analyze its own limitations. In this class, we'll study problems from various domains—geometry, algebra, analysis, number theory, computer science, and logic itself—and we'll explore which ones can or cannot be solved. The problems will all be relatively elementary (no specialized background will be required beyond a clear understanding of proofs and abstraction, obtained from 18.100 and one further course), but they will be fundamental to how we think about mathematics.

Whenever possible, we'll examine the original research papers. This is not always so easy, and we'll certainly fall back on modern expositions whenever the original papers are overly obscure or difficult to read, but it's important to understand where the ideas actually came from. Mathematics is not just a collection of facts, but also a story, so we need to understand both what and why.

- Course information
- Questionnaire
- Playing games with games: the Hypergame paradox, by William Zwicker
- Social processes and proofs of theorems and programs, by R. De Millo, R. Lipton, and A. J. Perlis
- On computable numbersm with an application to the Entscheidungsproblem, by Alan Turing (see also the correction)
- An unsolvable problem of elementary number theory, by Alonzo Church
- Recursively enumerable sets of positive integers and their decision problems, by Emil Post
- On non-computable functions, by Tibor Radó
- Universality of tag systems with P = 2, by John Cocke and Marvin Minsky
- Computability and recursion, by Robert Soare
- The complexity of theorem-proving procedures, by S. Cook
- Reducibility among combinatorial problems, by R. Karp
- Relativizations of the P=?NP Question, by T. Baker, J. Gill, and R. Solovay
- The completeness of the first-order functional calculus, by Leon Henkin
- Henkin's method and the completeness theorem, by Guram Bezhanishvili
- The suprise examination paradox and the second incompleteness theorem, by S. Kritchman and R. Raz
- Undecidability and nonperiodicity for tilings of the plane, by R. M. Robinson
- Hilbert's Tenth Problem is unsolvable, by M. Davis
- Proof of recursive unsolvability of Hilbert's Tenth Problem, by J. P. Jones and Y. V. Matijesevic
- The decision problem for exponential Diophantine equations, by M. Davis, H. Putnam, and J. Robinson
- Register machine proof of the theorem on exponential Diophantine representation of enumerable sets, by J. P. Jones and Y. V. Matijasevic
- Some undecidable problems involving elementary functions of a real variable, by D. Richardson