A History of the Theory of Types with Special Reference to Developments After the Second Edition of Principia Mathematica

Abstract

<p>This thesis traces the development of the theory of types from its origins in the early twentieth century through its various forms until the mid 1950's. Special attention is paid to the reception of this theory after the publication ofthe second edition of Whitehead and Russell's Principia Mathematica. We examine how the theory of types declined in influence over four decades. From being in the 1920s the dominant form of mathematical logic, by 1956 this theory had been abandoned as a foundation for mathematics. The use and modification of the theory by logicians such as Ramsey, Carnap, Church, Quine, Gode and Tarski is given particular attention. Finally, the view of the theory of types as a many-sorted first-order theory in the 1950's is discussed.</p> <p>It was the simple theory, as opposed to the ramified theory of types that was used almost exclusively during the years following the second edition of Principia. However, it is shown in this thesis that in the 1950's a revival of the ramified theory oftypes occurred. This revival of ramified-type theories coincided with the consideration of cumulative type hierarchies. This is most evident in the work ofHao Wang and John Myhill. The consideration of cumulative type-hierarchies altered the form of the theory of types in a substantial way. The theory was altered even more drastically by being changed from a many-sorted theory into a one-sorted theory. This fmal "standardization" of the theory of types in the mid 1950's made it not much different from first-order Zermelo-Fraenkel set-theory. The theory of types, whose developments are traced in this thesis, therefore lost its prominence as the foundation for mathematics and logic.</p>