Types in Logic and Mathematics before 1940

Kamareddine, Fairouz; Laan, Tdl; Nederpelt, RP Rob

doi:10.2307/2693964

Cited by 3 publications

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“…In [Nederpelt, 2002] a language WWT is developed with similar aims as our language MPL. So far the development of WWT has been focussed on the theory development: the formulation of the concepts and of the statements of the lemmas.…”

Section: Resultsmentioning

confidence: 99%

“…[Nederpelt, 2002] has a different approach: there the emphasis is on the ease of providing formalizations of mathematical definitions. At present it is mathematically not appealing to construct formal proofs.…”

Section: Towards An Interactive Mathematical Proof Mode*mentioning

confidence: 99%

“…A useful .>.-notation. [Kamareddine et ai, 2002] Fairouz Kamareddine, Twan Laan and Rob Nederpelt, "Types in mathematics and logic before 1940", Bulletin of Symbolic Logic, vol. [Kamareddine et ai, 2002] Fairouz Kamareddine, Twan Laan and Rob Nederpelt, "Types in mathematics and logic before 1940", Bulletin of Symbolic Logic, vol.…”

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confidence: 99%

“…Accepted papers from that call appear in [Kamareddine, 2002]. Accepted papers from that call appear in [Kamareddine, 2002].…”

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confidence: 99%

“…The article builds on an earlier formalization of RTT in [Kamareddine et aI, 2002]. Holmes attempts to remain close to Russell's formalisation whereas [Kamareddine et aI, 2002] follows the more modern style of type theory. Holmes attempts to capture those places accurately.…”

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confidence: 99%

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Section: Resultsmentioning

confidence: 99%

Section: Towards An Interactive Mathematical Proof Mode*mentioning

confidence: 99%

mentioning

confidence: 99%

“…Accepted papers from that call appear in [Kamareddine, 2002]. Accepted papers from that call appear in [Kamareddine, 2002].…”

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confidence: 99%

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confidence: 99%

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Polymorphic Type-Checking for the Ramified Theory of Types of Principia Mathematica

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2003

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A Constructive Examination of a Russell-Style Ramified Type Theory

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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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Types in Logic and Mathematics before 1940

Cited by 3 publications

References 0 publications

Thirty Five Years of Automating Mathematics

Thirty Five Years of Automating Mathematics

Polymorphic Type-Checking for the Ramified Theory of Types of Principia Mathematica

A Constructive Examination of a Russell-Style Ramified Type Theory

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