In this paper the theory of logical types will be examined, and certain departures from it will be suggested. Though the purpose of the paper is not primarily expository, an approach has been possible which presupposes no familiarity with special literature. Matters at variance with such an approach have been confined to appendices and footnotes.

In the early pages the logical paradoxes will be considered—an infinite series of them, of which Russell's paradox is the first. Then Russell's simple theory of types will be formulated, in adaptation to a minimal set of logical primitives: inclusion and abstraction. Two aspects of the theory will be distinguished: an ontological doctrine and a formal restriction. It will be found that by repudiating the former we can avoid certain unnatural effects of the type theory—notably the reduplication of logical constants from type to type, and the apparent dependence of finite arithmetic upon an axiom of infinity. But the formal restriction itself has unnatural effects, which survive, even in an aggravated form, after the type ontology has been dropped. A liberalization of the formal restriction will be proposed which removes the more irksome of these anomalies.

When the axioms of infinity and choice are added to the system of logic of the second edition of Whitehead and Russell's Principia mathematica, and when, as in the second edition, the axiom of reducibility is omitted (so that the ramified nature of Russell's theory of types, especially his distinction between “orders,” is not in effect obliterated), the resulting system will hereafter be referred to as “the ramified Principia.” It will be shown that a certain system S, slightly stronger than the ramified Principia, is consistent, so that the ramified Principia itself is consistent. From Gödel's theorem it will then follow that the sort of mathematical induction employed in the consistency proof of S cannot be adequately handled within S or within the ramified Principia.

The method of proving the consistency of S will be roughly as follows: A consistent non-constructive system S′ will be defined by means of induction with respect to a serial well-ordering of all the propositions of S. It will then be shown that every true proposition of S is a true proposition of S′, so that S must also be consistent.

1. Preliminary conventions.

1.1. Definition. By a “symbol” is to be understood any typographical expression which does not consist of a horizontal collinear sequence of expressions. Expressions formed by adding indices to either of the two upper or two lower corners of a symbol will also be considered symbols.

Consider a system of formal notations for ordinal numbers in the first and second number classes, with the following properties. Given a notation for an ordinal, it can be decided effectively whether the ordinal is zero, or the successor of an ordinal, or the limit of an increasing sequence of ordinals. In the second case, a notation for the preceding ordinal can be determined effectively. In the third case, notations for the ordinals of an increasing sequence of type ω with the given ordinal as limit can be determined effectively.

Are there systems of this sort which extend farthest into the second number class? When the conditions for the systems have been made precise, the question will be answered in the affirmative. There is an ordinal ω1 in the second number class such that there are systems of notations of the sort described which extend to all ordinals less than ω1, but none in which ω1 itself is assigned a notation.

1. An effective or constructive operation on the objects of an enumerable class is one for which a fixed set of instructions can be chosen such that, for each of the infinitely many objects (or n-tuples of objects), the operation can be completed by a finite process in accordance with the instructions. This notion is made exact by specifying the nature of the process and set of instructions. It appears possible to do so without loss of generality.