A study of substitution, using nominal techniques and Fraenkel-Mostowksi sets

Fraenkel-Mostowski (FM) set theory delivers a model of names and alpha-equivalence. This model, now generally called the ‘nominal’ model, delivers inductive datatypes of syntax with alpha-equivalence — rather than inductive datatypes of syntax, quotiented by alpha-equivalence. The treatment of names and alpha-equivalence extends to the entire sets universe. This has proven useful for developing ‘nominal’ theories of reasoning and programming on syntax with alpha-equivalence, because a sets universe includes elements representing functions, predicates, and behaviour. Often, we want names and alpha-equivalence to model capture-avoiding substitution. In this paper we show that FM set theory models capture-avoiding subsitution for names in much the same way as it models alpha-equivalence; as an operation valid for the entire sets universe which coincides with the usual (inductively defined) operation on inductive datatypes. In fact, more than one substitution action is possible (they all agree on sets representing syntax). We present two distinct substitution actions, making no judgement as to which one is ‘right’ — we suspect this question has the same status as asking whether classical or intuitionistic logic is ‘right’. We describe the actions in detail, and describe the overall design issues involved in creating any substitution action on a sets universe. Along the way, we think in new ways about the structure of elements of FM set theory. This leads us to some interesting mathematical concepts, including the notions of planes and crucial elements, which we also describe in detail.1