Philosophy 1880: Course Description

What Is Advanced Deductive Logic?

Philosophy 1880 is an introduction to the so-called `limitative' theorems concerning first-order logic. The most famous of these are the Gödel incompleteness theorems, but we shall also study Church's theorem, on the undecidability of first-order logic, and Tarski's theorem, on the undefinability of truth.

We will begin with what is known as "recursion theory" and develop a rigourous account of what it is for a function to be computable, or for a predicate (or property) to be decidable. We will meet concrete examples of uncomputable functions and then turn to the study of first-order logic itself and of its relation to computability. We will give a rigorous proof of the completeness of first-order logic, and see how this result can be understood in terms of computability.

We will then turn our attention to theories of arithemtic. This will first lead us to a proof of Church's theorem. Then we shall lay the groundwork for a proof of the first of the incompleteness theorems and then discuss the proof of the second incompleteness theorem, whose details we shall probably not have time to study.

Richard Heck Department of Philosophy Brown University