Constructive Examination of a Russell-style Ramified Type Theory

In this paper we examine the natural interpretation of a ramified type hierarchy into Martin-Löf type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of Russell’s reducibility axiom are valid, namely functional reducibility. This is sufficient to make the type hierarchy usable for development of constructive mathematics. We present a ramified type theory suitable for this purpose. One may regard the results of this paper as an alternative solution to the problems of Russell’s theory, which avoids impredicativity, but instead imposes constructive logic. Mathematics Subject Classification (2010): 03B15, 03F35, 03F50 Russell introduced with his ramified type theory a distinction between different levels of propositions in order to solve logical paradoxes, notably the Liar Paradox and the paradox he discovered in Frege’s system (Russell 1908). To be able to carry out certain mathematical constructions, e.g. the real number system, he was then compelled to introduce the reducibility axiom. This had however the effect of collapsing the ramification, from an extensional point of view, and thus making the system impredicative. The original Russell theory is not quite up to modern standards of presentation of a formal system: a treatment of substitution is lacking. In the article by Kamareddine, Laan and Nederpelt (2002) however a modern reconstruction of Russell’s type theory using lambda-calculus notation is given. We refer to their article for further background and history. In this paper we shall present an intuitionistic version of ramified type theory IRTT. By employing a restricted form of reducibility it can be shown to be predicatively acceptable. This axiom, called the Functional Reducibility Axiom (FR), reduce type levels only of total functional relations. The axiom is enough to handle the problem of proliferation of levels of real numbers encountered in Russell’s original theory. It is essential that it is based on intuitionistic logic as (FR) imply the full reducibility principle with classical logic. The system IRTT is demonstrated to be predicative by interpreting it in a subsystem of Martin-Löf type theory (1984), a system itself predicative in the proof-theoretic sense of Feferman and Schütte (Feferman 1982). One may regard the results of this paper as an alternative solution to the problems of Russell’s theory which avoids impredicativity, but instead imposes constructive logic. The interpretation is carried out in Sections 2 and 3. In Section 4, we see how universal set constructions useful for e.g. formalizing real numbers can be carried out Date: April 18, 2017. The author was supported by a grant from the Swedish Research Council (VR). Author’s address: Department of Mathematics, Stockholm University, 106 91 Stockholm. Email: palmgren[at]math.su.se. 1