In combinatory logic there is a system of objects which intuitively represent functions, and a binary relation between these objects, which represents the process of evaluating the result of applying a function to an argument. (This is explained fully in [1].) From this relation called weak reduction, “≥,” an equivalence relation is defined by saying that X is weakly equivalent to Y if and only if there exist n (with 0 ≤ n) and X0,…,Xη such that

It turns out that equivalent objects represent the same function, but two objects representing the same function need not be equivalent.

We give a short proof of the theorem that, assuming the axiom of choice, every set can be well-ordered. The proof makes direct use of neither induction nor ordinals.

Let S be any given set and its power-set. Let be a choice function, i.e. we denote Ζ — {γΖ} by Ζ'.

Methods are being developed for treating questions of decidability in fewer than Ω steps, Ω being a regular nondenumerable cardinal. In this paper we consider the set-theoretical predicate Taut(x), “x is a tautology,” taken in the infinitary sense. In case x is hereditarily finite there is no question that it is decidable in finitely many steps. But what if x is infinite? We are not assuming that x is in any way constructive or even that the propositional formulas can be well-ordered, so it is not appropriate to treat this predicate as one of natural numbers or even as one of ordinals.

In his paper [3] Hindley shows that strong reduction in combinatory logic (see [1] for the basic discussion of this notion) cannot be axiomatized with a finite number of axiom schemes, and then he presents an infinite system of axiom schemes for strong reduction. Hindley lists one axiom and six axiom schemes, together with a method (clause (viii) below) for generating further axiom schemes from these. The results of this note are that different applications of clause (viii) yield different axiom schemes, and that the property of being an axiom is decidable.