A modern elaboration of the ramified theory of types

Laan, Tdl; Nederpelt, RP Rob

doi:10.1007/bf00370835

Cited by 7 publications

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“…Remark 2.2. The ramified types of Russell are-according to the modern elaboration of Laan and Nederpelt [6]-given by the following inductive definition. Each ramified type has the form t n where n is a natural number indicating the order of the type.…”

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A Constructive Examination of a Russell-Style Ramified Type Theory

Palmgren¹

2018

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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. The intuitionistic ramified type theory introduced here, also suggests that there is a natural associated notion of predicative elementary topos. Mathematics Subject Classification (2010): 03B15, 03F35, 03F50Russell 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 presented. 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 (Kamareddine et al. 2002, pp. 231 -232). It is essential that theory is based on intuitionistic logic, as (FR) imply the full reducibility principle using classical logic. The system IRTT is demonstrated to be predicative by interpreting it in a subsystem of Martin-Löf type theory (Martin-Löf 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.

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“…Conversely, suppose G f ( a, b ), so that f( a, p ) = V b, q for some p : X (a) and q : Y (b). By(6) we have F ( 1 ( a, p ), 1 (f( a, p )) ). Thus since F is extensional, F ( a, b ).…”

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A Constructive Examination of a Russell-Style Ramified Type Theory

Palmgren¹

2018

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“…Whitehead and Russell present their theory, the Ramified Type Theory (rtt), in an informal way. Several rough descriptions of this theory have been given in the literature (see for instance [14,15,37,59]) but we present a formalisation of rtt given in [48] that is directly based on the presentation of rtt in Whitehead and Russell's Principia Mathematica ( [71], 1910-12). The construction of this formalisation is not a simple task.…”

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“…Though there are many descriptions of rtt available in the literature (like [14,15,37,59] and Section 27 of [65]), none of these descriptions presents a formalisation that is both accurate and as close as possible to the ideas of the Principia. [47,48] fill up this gap in the literature. In this section, we review the formalisation of [47, 48] explaining how it faithfully represents the intentions of Russell and Whitehead.…”

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“…Our article is not only intended to introduce the prehistory of type theory (up to 1910) and its development between Principia Mathematica ( [71], [1910][1911][1912]) and Church's simply typed -calculus of 1940, but also we will present Russell's ramified theory of types and the simple theory of types due to Ramsey, in a modern setting. 4 The presentation of the ramified theory of types in a modern setting was already given in [48] but there, neither the deramification nor the connection of the ramified type theory with that of Church's simple theory of types was given. Similarly, although these theories have already been described in a modern framework, the relation between the modern description and the original system has not always been made clear.…”

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Types in Logic and Mathematics Before 1940

Kamareddine¹,

Laan²,

Nederpelt³

2002

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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 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) and we finish by comparing RTT, STT and λ→C.

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