Mr. Shen Yuting, in this Journal, vol. 18, no. 2 (June, 1953), stated a new paradox of intuitive set-theory. This paradox involves what Mr. Yuting calls the class of all grounded classes, that is, the family of all classes a for which there is no infinite sequence b such that … ϵ bn ϵ … ϵ b2ϵb1 ϵ a.

Now it is possible to state this paradox without employing any complex set-theoretical notions (like those of a natural number or an infinite sequence). For let a class x be called regular if and only if (k)(x ϵ k ⊃ (∃y)(y ϵ k · ~(∃z)(z ϵ k · z ϵ y))). Let Reg be the class of all regular classes. I shall show that Reg is neither regular nor non-regular.

Suppose, on the one hand, that Reg is regular. Then Reg ϵ Reg. Now Reg ϵ ẑ(z = Reg). Therefore, since Reg is regular, there is a y such that y ϵ ẑ(z = Reg) · ~(∃z)(z ϵ z(z = Reg) · z ϵ y). Hence ~(∃z)(z ϵ ẑ(z = Reg) · z ϵ Reg). But there is a z (namely Reg) such that z ϵ ẑ(z = Reg) · z ϵ Reg.

On the other hand, suppose that Reg is not regular. Then, for some k, Reg ϵ k · [1] (y)(y ϵ k ⊃ (∃z)(z ϵ k · z ϵ y)). It follows that, for some z, z ϵ k · z ϵ Reg. But this implies that (ϵy)(y ϵ k · ~(ϵw)(w ϵ k · w ϵ y)), which contradicts [1].

It can easily be shown, with the aid of the axiom of choice, that the regular classes are just Mr. Yuting's grounded classes.

The purpose of this paper is to present and justify a simple proof procedure for quantification theory. The procedure will take the form of a method for proving a quantificational schema to be inconsistent, i.e., satisfiable in no non-empty universe. But it serves equally for proving validity, since we can show a schema valid by showing its negation inconsistent.

Method A, as I shall call it, will appear first, followed by a more practical adaptation which I shall call B. The soundness and completeness of A will be established, and the equivalence of A and B. Method A, as will appear, is not new.

The reader need be conversant with little more than the fairly conventional use (as in [8]) of such terms as ‘quantificational schema’, ‘interpretation’, ‘valid’, ‘consistent’, ‘prenex’, and my notation (as in [7]) of quasi-quotation.

In a recent paper (here referred to as “IDP”) the writer outlined a decision procedure for Lewis's system S4 of modal logic. One of the clauses in definition 3.1 of IDP requires correction. Clause II of 3.1 (2) should read as follows.

II. Some constituent of the form ◊β, of degree n1 ≤ n, has the value T in Row (i), and some constituents of the forms ◊δ1, … ◊δh, and ◊η1, …, ◊ηm, all have the value F in Row (i) (h ≥ 0, m ≥ 0, h+m ≥ 1), where β → (δ1 ∨ … ∨ δh ∨ ◊η1 ∨ … ∨ ◊ηm) is an (n1 − 1)-tautology of S4.

This change is required in order to carry out the proof of Metathcorcm 3.19. In particular, the change guarantees the following. If the expression η of the second paragraph of 3.20 is of degree n − 1, then the antecedent λ of formula ζ on page 210 of IDP is also of degree n − 1; and consequently the formula ζ of 3.19 is of degree n (since ◊λ is a constituent of ζ). (If we fail to make the correction, then it might be the case that both ζ and (3) are of degree n − 1, in which case Row (i) would not satisfy clause II as originally stated, contrary to the claim at the end of 3.20.) The proofs for the remaining cases of 3.20 can then be carried out, using the revised clause II, in the way originally indicated.

The proof of Metatheorem 3.2 requires only trivial corrections for case 2.

Cantor's second ordinal number class is perhaps the simplest example of a set of mathematical objects which cannot all be named in one language. In this paper we shall investigate a system of names for a segment of the first and second number classes in relation to decision problems. The system, except for one minor difference, is the one studied by Markwald in [12]. In our system ordinals are named by natural numbers from a set W via recursive well-orderings of subsets of the natural numbers.

The decision problems will be related to the hyperarithmetical hierarchy of Davis [2], [3] and Kleene [8]. This hierarchy is indexed by ordinal notations from Kleene's system S3 [4], [6], [9], in which ordinals are named by natural numbers from a set O, partially well-ordered ([12] p. 138) by a relation a≤Ob; O and ≤O are defined inductively by applications of the successor and limit operations. As results of this investigation, we shall (i) answer negatively Markwald's question [12] Theorem 12 whether his set “W” is arithmetical by showing that it is not even hyperarithmetical, (ii) obtain a new proof of the main result of Kleene [10] that every predicate expressible in both the one-function-quantifier forms of [8] is recursive in Hα for some aεO, (iii) answer affirmatively the question raised by Davis [2], [3] whether all the Church-Kleene constructive ordinals are uniqueness ordinals, and (iv) solve the function-quantifier analog of Post's problem [15]. Strong use will be made of the well-orderings that can be constructed from one-function-quantifier predicates as in [9].