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Switch to: Citations
References in:
Weyl Reexamined: “Das Kontinuum” 100 Years Later
Bulletin of Symbolic Logic 26 (1):26-79 (2020)
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This paper is divided into two parts. Part I provides a resumé of the evolution of the notion of predicativity. Part II describes our own work on the subject.Part I§1. Conceptions of sets.Statements about sets lie at the heart of most modern attempts to systematize all (or, at least, all known) mathematics. Technical and philosophical discussions concerning such systematizations and the underlying conceptions have thus occupied a considerable portion of the literature on the foundations of mathematics. |
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Dans le Continu, Hermann Weyl donne une nouvelle assise aux notions d’ensemble et de fonction, pour assurer aux mathématiques leur applicabilité à la physique, et résoudre ainsi le problème du continu. Les notions introduites, éloignées de la théorie des ensembles, prêtent à confusion et à multiples interprétations.Nous nous proposons d’éclairer le sens du déplacement que Weyl opère dans ces notions. Nous présentons une synthèse des thèses épistémologiques soutenues dans Le Continu et résolvons certains problèmes interprétatifs. Par une approche comparative, nous (...) |
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In these selected essays, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the past century: Brouwer, Hilbert, Bernays, Weyl, Gödel, Russell, Quine, Putnam, Wang, and Tait. |
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We suggest a new framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory P ZF which resembles ZF , and is suitable for mechanization. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. The language of P ZF is type-free, and it reflects real mathematical practice in making an extensive use of statically (...) |
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Two more or less standard methods exist for the systematic, logical construction of classical mathematics, the so-called theory of types, due in the main to Russell, and the Zermelo axiomatic set theory. In systems based upon either of these, the connective of membership, “ε”, plays a fundamental role. Usually although not always it figures as a primitive or undefined symbol.Following the familiar simplification of Russell's theory, let us mean by alogical typein the strict sense any one of the following: (i) (...) |
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This chapter is a detailed study of predicativity in mathematics. It presents a number of historical versions predicativity requirements, looking for unifying ideas. The further development of the notions and requirements up to the present is traced, articulating connections among the different ideas. One underlying theme of the chapter is the motivations for the various requirements for rejecting impredicativity and the various ways of stating the requirement. |
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Many efforts have been made in recent years to construct formal systems for mechanizing general mathematical reasoning. Most of these systems are based on logics which are stronger than first-order logic. However, there are good reasons to avoid using full second-order logic for this task. In this work we investigate a logic which is intermediate between FOL and SOL, and seems to be a particularly attractive alternative to both: ancestral logic. This is the logic which is obtained from FOL by (...) |
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A logic-enriched type theory is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named and , which we claim correspond closely to the classical predicative systems of second order arithmetic and . We justify this claim by translating each second order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches (...) |
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