Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.