Intuitionistic Sets and Ordinals

Transitive extensional well founded relations provide an intuitionistic notion of ordinals which admits transfinite induction. However these ordinals are not directed and their successor operation is poorly behaved, leading to problems of functoriality. We show how to make the successor monotone by introducing plumpness, which strengthens transitivity. This clarifes the traditional development of successors and unions, making it intuitionistic; even the (classical) proof of trichotomy is made simpler. The definition is, however, recursive, and, as their name suggests, the plump ordinals grow very rapidly. Directedness must be defined hereditarily . It is orthogonal to the other four conditions, and the lower powerdomain construction is shown to be the universal way of imposing it. We treat ordinals as order-types, and develop a corresponding set theory similar to Osius’ transitive set objects. This presents Mostowski’s theorem as a reflection of categories, and set-theoretic union is a corollary of the adjoint functor theorem. Mostowski’s theorem and the rank for some of the notions of ordinal are formulated and proved without the axiom of replacement, but this seems to be unavoidable for the plump rank. The comparison between sets and toposes is developed as far as the identification of replacement with completeness and there are some suggestions for further work in this area. Each notion of set or ordinal defines a free algebra for one of the theories discussed by Joyal and Moerdijk, namely joins of a family of arities together with an operation s satisfying conditions such as x ≤ sx, monotonicity or s(x ∨ y) ≤ sx ∨ sy. Finally we discuss the fixed point theorem for a monotone endofunction s of a poset with least element and directed joins. This may be proved under each of a variety of additional hypotheses. We explain why it is unlikely that any notion of ordinal obeying the induction scheme for arbitrary predicates will prove the pure result. Presented at Category Theory and Computer Science 5, Amsterdam, September 1993. Published in the Journal of Symbolic Logic 61 (1996) 705–744. 1 Well founded induction Transfinite induction is very widely and readily used in classical mathematics. In its extreme form — the Procrustean enumeration of elements of arbitrary sets known as the “well” ordering principle — it is equivalent to choice and is irreconcilable with constructivity. However ordinals are also used in proof theory and infinitary algebra to extend the iterative constructions used in finitary cases, so it would be of great benefit to include them in an intuitionistic categorical account. Georg Cantor [4] defined ordinals as well founded relations which satisfy the trichotomy law, (x ≺ y) ∨ (x = y) ∨ (y ≺ x), but to show that unions of ordinals have this property depends on excluded middle. However trichotomy is provable (classically, Proposition 4.8) if ordinals are re-defined as “transitive sets 1I have been unable to identify who first discovered this (non-trivial) fact, which certainly deserves credit.