A New Approach to Abstract Syntax Involving Binders

The Fraenkel-Mostowski permutation model of set theory with atoms (FM-sets) can serve as the semantic basis of meta-logics for specifying and reasoning about formal systems involving name binding, -conversion, capture avoiding substitution, and so on. We show that in FM-set theory one can express statements quantifying over ‘fresh’ names and we use this to give a novel set-theoretic interpretation of name abstraction. Inductively defined FM-sets involving this name-abstraction set former (together with cartesian product and disjoint union) can correctly encode object-level syntax modulo -conversion. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated notion of structural recursion for defining syntax-manipulating functions (such as capture avoiding substitution, set of free variables, etc) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice.