Towards a Unified Treatment of Induction, I: The General Recursion Theorem

The recursive construction of a function f : A → Θ consists, paradigmatically, of finding a functor T and maps α : A→ TA and θ : TΘ→ Θ such that f = α ; Tf ; θ. The role of the functor T is to marshall the recursive sub-arguments, and apply the function f to them in parallel. This equation is called partial correctness of the recursive program, because we have also to show that it terminates, i.e. that the recursion (coded by α) is well founded. This may be done by finding another map g : A→ N , called a loop variant, where N is some standard well founded srtucture such as the natural numbers or ordinals. In set theory the functor T is the covariant powerset; in the study of the free algebra for a free theory Ω (such as in proof theory) it is the polynomial Σr∈Ω(−), and it is often something very crude. We identify the properties of the category of sets needed to prove the general recursion theorem, that these data suffice to define f uniquely. For any pullback-preserving functor T , a structure similar to the von Neumann hierarchy is developed which analyses the free T algebra if it exists, or deputises for it otherwise. There is considerable latitude in the choice of ambient category, the functor T and the class of predicates admissible in the induction scheme. Free algebras, set theory, the familiar ordinals and novel forms of them which have arisen in theoretical computer science are treated in a uniform fashion. The central idea in the paper is a categorical definition of well founded coalgebra α : A . TA, namely that any pullback diagram of the form