Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λ-calculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced

In Russell's Ramified Theory of Types rtt, two hierarchical concepts dominate: orders and types. The use of orders has as a consequence that the logic part of rtt is predicative. The concept of order however, is almost dead since Ramsey eliminated it from rtt. This is why we find Church's simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than rtt. Despite the disappearance of orders which have a strong correlation with predicativity, predicative logic still plays an influential role in Computer Science. An important example is the proof checker Nuprl, which is based on Martin-Löf's Type Theory which uses type universes. Those type universes, and also degrees of expressions in Automath, are closely related to orders. In this paper, we show that orders have not disappeared from modern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl. The novelty of our paper lies in: 1) a modest revival of Russell's orders, 1 2) the placing of the historical system rtt underlying the famous Principia Mathematica in a context with a modern system of computer mathematics (Nuprl) and modern type theories (Martin-Löf's type theory and PTSs), and 3) the presentation of a complex type system (Nuprl) as a simple and compact PTS.

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While unification in the simple theory of types (a.k.a. higher-order logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable unification problem even at order 2. However, the decidability result for the pure subsystem indicates that unification terminates more often than general higher-order unification. We present an application to ACA 0 and other expressive subsystems of second-order Peano arithmetic.

statement of his theory of logical types.This simple version of the theory is designed to block the reasoning that leads to the paradox of the Russell class. But Russell notes immediately that new problems arise.The problems culminate in the paradox of propositions.This is a problem that seems to run in exact parallel to the paradox of the Russell class.It seemed therefore desirable to Russell that a single solution to both paradoxes be found.Since the simple theory of types (ST) does not offer such a solution it is commonly believed that the paradox of propositions was Russell’s principal motive— at least at the time when he had just finished writing the Principles—for searching for and eventually formulating a ramified theory of types (RT). In the next section I shall present the very first version of ST as it occurs in the Principles.I shall explain in which direction Russell was seeking for a solution to the paradox of propositions which would run in parallel to his solution to the class paradox. Next I turn to the Russell-Frege correspondence of 1902 and 1903.Apart