Abstract: Given the lack of progress on complexity lower bounds, it is natural to ask whether they are hard to prove, in some formal sense. I will begin by briefly describing the classical incompleteness results of Godel and Chaitin, and posing the question for whether there are analogues of these results in complexity theory.

I will then introduce the finitistic framework of propositional proof complexity, where we are interested in the existence of polynomial size proofs verifiable in polynomial time. I will explain what it means to prove circuit complexity or proof complexity lower bounds in this framework. Finally, I will describe a strong complexity conjecture for which it can be shown unconditionally that there are no feasible propositional proofs, in a certain technical sense.

(Based on joint work with Jan Pich).