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 states that no consistent axiomatic theory of sufficient strength and expressive power proves its own consistency. 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. Part of our purpose in this course will be simply to understand what the theorems really say.
The first incompleteness theorem underwent significant refinement in the years following Gödel's original publication. What is now its 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.)
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 get the sets of proofs and theorems right, but that seem to allow theories to prove their own consistency, which they are not supposed to be able to do. We will look at some early work on this issue and then, if we have time, look at more recent work that seeks to refine our understanding of the insight the second incompleteness theorem seems to give us.