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.