Iterative Conceptions of Set
Iterative Conceptions of Set

Author: Neil Barton

Publisher: Cambridge University Press

Published: 2024-06-30

Total Pages: 122

ISBN-13: 1009227254

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Many philosophers are aware of the paradoxes of set theory (e.g. Russell's paradox). For many people, these were solved by the iterative conception of set which holds that sets are formed in stages by collecting sets available at previous stages. This Element will examine possibilities for articulating this solution. In particular, the author argues that there are different kinds of iterative conception, and it's open which of them (if any) is the best. Along the way, the author hopes to make some of the underlying mathematical and philosophical ideas behind tricky bits of the philosophy of set theory clear for philosophers more widely and make their relationships to some other questions in philosophy perspicuous.

Conceptions of Set and the Foundations of Mathematics
Conceptions of Set and the Foundations of Mathematics

Author: Luca Incurvati

Publisher: Cambridge University Press

Published: 2020-01-23

Total Pages: 255

ISBN-13: 1108758355

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Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics.

Set Theory and its Philosophy
Set Theory and its Philosophy

Author: Michael Potter

Publisher: Clarendon Press

Published: 2004-01-15

Total Pages: 362

ISBN-13: 0191556432

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Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.

Logic, Meaning and Computation
Logic, Meaning and Computation

Author: Alonzo Church

Publisher: Springer Science & Business Media

Published: 2001

Total Pages: 664

ISBN-13: 9781402001413

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This volume began as a remembrance of Alonzo Church while he was still with us and is now finally complete. It contains papers by many well-known scholars, most of whom have been directly influenced by Church's own work. Often the emphasis is on foundational issues in logic, mathematics, computation, and philosophy - as was the case with Church's contributions, now universally recognized as having been of profound fundamental significance in those areas. The volume will be of interest to logicians, computer scientists, philosophers, and linguists. The contributions concern classical first-order logic, higher-order logic, non-classical theories of implication, set theories with universal sets, the logical and semantical paradoxes, the lambda-calculus, especially as it is used in computation, philosophical issues about meaning and ontology in the abstract sciences and in natural language, and much else. The material will be accessible to specialists in these areas and to advanced graduate students in the respective fields.

Logic in Linguistics
Logic in Linguistics

Author: Jens Allwood

Publisher: Cambridge University Press

Published: 1977-09-15

Total Pages: 200

ISBN-13: 9780521291743

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The authors offer a clear, succinct and basic introduction to set theory and formal logic for linguists.

From Mathematics to Philosophy (Routledge Revivals)
From Mathematics to Philosophy (Routledge Revivals)

Author: Hao Wang

Publisher: Routledge

Published: 2016-06-10

Total Pages: 445

ISBN-13: 1134884338

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First published in 1974. Despite the tendency of contemporary analytic philosophy to put logic and mathematics at a central position, the author argues it failed to appreciate or account for their rich content. Through discussions of such mathematical concepts as number, the continuum, set, proof and mechanical procedure, the author provides an introduction to the philosophy of mathematics and an internal criticism of the then current academic philosophy. The material presented is also an illustration of a new, more general method of approach called substantial factualism which the author asserts allows for the development of a more comprehensive philosophical position by not trivialising or distorting substantial facts of human knowledge.