Substitution for Fraenkel-Mostowski foundations

A fundamental and unanalysed logical concept is substitution. This seemingly innocuous operation— substituting a variable for a term or valuating a variable to an element of a domain — is hard to characterise other than by concrete constructions. It is widely viewed as a technicality to be dispensed with on the way to studying other things. Discussions of computer science foundations, and of the philosophy of logic, have largely ignored it. We show that Fraenkel-Mostowski set theory gives a model of variables and substitution as constructions on sets. Thus models of variables and substitution are exhibited as constructions in a foundational universe, just like models of arithmetic (the ordinals) and other mathematical entities. The door is open for classes of denotations in which variables, substitution, and evaluations are constructed directly in sets and studied independently of syntax, in ways which would previously have not been possible.