A Correspondence between Martin-Löf Type Theory, the Ramified Theory of Types and Pure Type Systems

Abstract

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-Lof'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, (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-Lof's type theory and PTSs), and (3) the presentation of a complex type system (Nuprl) as a simple and compact PTS.