Schröder's anticipation of the simple theory of types

Abstract

The attempt to exclude the well-known pardoxes of set theory from a system of formal logic, without at the same time rendering the system so weak as to be inadequate for the purposes of mathematics, may be made in various ways. Currently the most favored methods are the Zermelo axiomatic set theory and the simple theory of types. As is well known, the simple theory of types may be described in the following terms. A particular domain, to be called the domain of individuals is selected; this may be any domain within very wide limits, but in any particular system must be treated as fixed. Classes and relations (and functions, if the system provides for such) are then classified into a hierarchy of types. For simplicity in the description of this hierarchy it is convenient to think of relations (and functions) as defined in terms of classes following a suggestion first made by Wiener in 1914. On this basis, the first type is composed of the individuals, the second type of classes of individuals, the third type of classes of classes of individuals, and so on. The restriction imposed by the simple theory of types is that the members of a class which belongs to the type n +1 must all belong to the type n. There is no provision for classes which do not belong to a type: classes not obeying the restriction just stated are regarded as non? existent, and names for such classes are excluded from the system. And even to say or write a e b (that a is member of the class b) is excluded as not well-formed unless the variables or constants a, b show by their syntactical form that, for some n, they refer to types n and n + 1 respectively. (Of course, the foregoing statement cannot be made within the system based on the simple theory of types which is being described, but only within some stronger system, e.g., one employing transfinite types. For the purpose of the formal construction of the system it is not