In 1931, the great German logician Kurt Gödel published his paper "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I", which contained his celebrated incompleteness theorems. The first of these states that, in any consistent axiomatic theory of sufficient strength and expressive power, there will always be a statement that is undecidable by the theory, in the sense that the theory neither proves nor refutes this statement. The second incompleteness theorem gives us a particularly interesting example of such a statement: No consistent axiomatic theory of sufficient strength and expressive power proves that it is consistent. It is an intriguing corollary of the second incompleteness theorem that there are consistent theories that prove their own inconsistency (falsely, of course, from which it follows that the theories themselves are false).
These two results, and the techniques used in their proof, have had a major influence on the development of mathematical logic. Much of what we do in this course will be simply to study what these two theorems really say.
The first incompleteness theorem underwent significant refinement in the years following Gödel's original publication. What is now the standard treatment did not really emerge until work by Tarski, Mostowski, and Robinson in the early 1950s. We will read their book, in part as a way of becoming familiar with the important notion of "interpretation", which is essential for stating the incompleteness theorems in a general way. (Note that this is a different notion from the semantic notion of interpretation that is introduced in Phil 0540 or Phil 1630.)
Our main focus, though, will be on the second incompleteness theorem: the one concerned with consistency. This result is, in some ways, puzzling. For it turns out that it is extremely sensitive to questions about how notions like proof and theory are formulated. There are ways of defining these notions that are extensionally correct, in the sense that they define the right sets (of proofs and of theorems), but, if one uses those definitions, then it turns out that many theories can prove their own consistency, which they are not supposed to be able to do; indeed, the proofs are trivial. We will look at Solomon Feferman's famous work on this issue at the end of the course.