A list of topics by lecture is available in the calendar listed below.
Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Description
We’ll begin with an introduction to the theory of computability, then proceed to a detailed study of its most illustrious result: Kurt Gödel’s theorem that, for any system of true arithmetical statements we might propose as an axiomatic basis for proving truths of arithmetic, there will be some arithmetical statements that we can recognize as true even though they don’t follow from the system of axioms. In my opinion, which is widely shared, this is the most important single result in the entire history of logic, important not only on its own right but for the many applications of the technique by which it’s proved. We’ll discuss some of these applications, among them: Church’s theorem that there is no algorithm for deciding when a formula is valid in the predicate calculus; Tarski’s theorem that the set of true sentence of a language isn’t definable within that language; and Gödel’s second incompleteness theorem, which says that no consistent system of axioms can prove its own consistency.
The text for the course will be lecture notes.
Grading
There will be homework assignments every week or two, and they will be the basis for your grade in the course. I would encourage you to work together on the homework, but when you finally sit down to write up your answers, you should work by yourself without looking at anyone else’s efforts.
Calendar
| LEC # | TOPICS | KEY DATES |
|---|---|---|
| 1 | Why Study Computability? | |
| 2-3 | Key Computability Concepts | |
| 4 | The Language of Arithmetic | |
| 5 | Church-Turing Thesis | |
| 6 | Nonstandard Models of Arithmetic | |
| 7 | Gödel Numbering | Homework 1 due |
| 8 | Robinson’s Arithmetic | Homework 2 due |
| 9-10 | Coding Proofs | |
| 11-12 | Peano Arithmetic | Homework 3 due in Lec #12 |
| 13-14 | Self-Reference Lemma | |
| 15-16 | First Incompleteness Theorem | Homework 4 due in Lec #16 |
| 17 | Interpretations | |
| 18 | Tarski’s Theory of Truth | Homework 5 due |
| 19 |
Gödel, Mechanism, and Mind Articles by Lucas and Benacerraf |
|
| 20-21 | Second Incompleteness Theorem | Homework 6 due in Lec #21 |
| 22 | Introduction to Modal Logic | |
| 23-24 | Provability Logic | Homework 7 due in Lec #23 |
| 25 | Defining Exponentiation | Homework 8 due |
