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A Logical Foundation for Potentialist Set Theory

A Logical Foundation for Potentialist Set Theory
Author: Sharon Berry
Publisher: Cambridge University Press
Total Pages: 250
Release: 2022-02-17
Genre: Science
ISBN: 1108998852
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In many ways set theory lies at the heart of modern mathematics, and it does powerful work both philosophical and mathematical – as a foundation for the subject. However, certain philosophical problems raise serious doubts about our acceptance of the axioms of set theory. In a detailed and original reassessment of these axioms, Sharon Berry uses a potentialist (as opposed to actualist) approach to develop a unified determinate conception of set-theoretic truth that vindicates many of our intuitive expectations regarding set theory. Berry further defends her approach against a number of possible objections, and she shows how a notion of logical possibility that is useful in formulating Potentialist set theory connects in important ways with philosophy of language, metametaphysics and philosophy of science. Her book will appeal to readers with interests in the philosophy of set theory, modal logic, and the role of mathematics in the sciences.

A Logical Foundation for Potentialist Set Theory

A Logical Foundation for Potentialist Set Theory
Author: Sharon Berry
Publisher: Cambridge University Press
Total Pages: 249
Release: 2022-02-17
Genre: Science
ISBN: 1108834310
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A new approach to the standard axioms of set theory, relating the theory to the philosophy of science and metametaphysics.

Conceptions of Set and the Foundations of Mathematics

Conceptions of Set and the Foundations of Mathematics
Author: Luca Incurvati
Publisher: Cambridge University Press
Total Pages: 255
Release: 2020-01-23
Genre: History
ISBN: 1108497829
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Presents a detailed and critical examination of the available conceptions of set and proposes a novel version.

The Routledge Handbook of Modality

The Routledge Handbook of Modality
Author: Otávio Bueno
Publisher: Routledge
Total Pages: 432
Release: 2020-12-29
Genre: Philosophy
ISBN: 1317585283
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Modality - the question of what is possible and what is necessary - is a fundamental area of philosophy and philosophical research. The Routledge Handbook of Modality is an outstanding reference source to the key topics, problems and debates in this exciting subject and is the first collection of its kind. Comprising thirty-five chapters by a team of international contributors the Handbook is divided into seven clear parts: worlds and modality essentialism, ontological dependence, and modality modal anti-realism epistemology of modality modality in science modality in logic and mathematics modality in the history of philosophy. Within these sections the central issues, debates and problems are examined, including possible worlds, essentialism, counterfactuals, ontological dependence, modal fictionalism, deflationism, the integration challenge, conceivability, a priori knowledge, laws of nature, natural kinds, and logical necessity. The Routledge Handbook of Modality is essential reading for students and researchers in epistemology, metaphysics and philosophy of language. It will also be very useful for those in related fields in philosophy such as philosophy of mathematics, logic and philosophy of science.

Set Theory

Set Theory
Author: Andras Hajnal
Publisher: Cambridge University Press
Total Pages: 332
Release: 1999-11-11
Genre: Mathematics
ISBN: 9780521596671
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This is a classic introduction to set theory in three parts. The first part gives a general introduction to set theory, suitable for undergraduates; complete proofs are given and no background in logic is required. Exercises are included, and the more difficult ones are supplied with hints. An appendix to the first part gives a more formal foundation to axiomatic set theory, supplementing the intuitive introduction given in the first part. The final part gives an introduction to modern tools of combinatorial set theory. This part contains enough material for a graduate course of one or two semesters. The subjects discussed include stationary sets, delta systems, partition relations, set mappings, measurable and real-valued measurable cardinals. Two sections give an introduction to modern results on exponentiation of singular cardinals, and certain deeper aspects of the topics are developed in advanced problems.

Set Theory

Set Theory
Author: Kenneth Kunen
Publisher:
Total Pages: 313
Release: 2006
Genre: Axiomatic set theory
ISBN: 9780444564023
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"Studies in Logic and the Foundations of Mathematics, Volume 102: Set Theory: An Introduction to Independence Proofs offers an introduction to relative consistency proofs in axiomatic set theory, including combinatorics, sets, trees, and forcing. The book first tackles the foundations of set theory and infinitary combinatorics. Discussions focus on the Suslin problem, Martin's axiom, almost disjoint and quasi-disjoint sets, trees, extensionality and comprehension, relations, functions, and well-ordering, ordinals, cardinals, and real numbers. The manuscript then ponders on well-founded sets and easy consistency proofs, including relativization, absoluteness, reflection theorems, properties of well-founded sets, and induction and recursion on well-founded relations. The publication examines constructible sets, forcing, and iterated forcing. Topics include Easton forcing, general iterated forcing, Cohen model, forcing with partial functions of larger cardinality, forcing with finite partial functions, and general extensions. The manuscript is a dependable source of information for mathematicians and researchers interested in set theory" -- Provided by publisher.

Mathematics and Its Logics

Mathematics and Its Logics
Author: Geoffrey Hellman
Publisher: Cambridge University Press
Total Pages: 296
Release: 2021-02-04
Genre: Science
ISBN: 1316999602
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In these essays Geoffrey Hellman presents a strong case for a healthy pluralism in mathematics and its logics, supporting peaceful coexistence despite what appear to be contradictions between different systems, and positing different frameworks serving different legitimate purposes. The essays refine and extend Hellman's modal-structuralist account of mathematics, developing a height-potentialist view of higher set theory which recognizes indefinite extendability of models and stages at which sets occur. In the first of three new essays written for this volume, Hellman shows how extendability can be deployed to derive the axiom of Infinity and that of Replacement, improving on earlier accounts; he also shows how extendability leads to attractive, novel resolutions of the set-theoretic paradoxes. Other essays explore advantages and limitations of restrictive systems - nominalist, predicativist, and constructivist. Also included are two essays, with Solomon Feferman, on predicative foundations of arithmetic.

The Construction of Logical Space

The Construction of Logical Space
Author: Agustín Rayo
Publisher:
Total Pages: 241
Release: 2013-06-27
Genre: Mathematics
ISBN: 0199662622
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Our conception of logical space is the set of distinctions we use to navigate the world. Agustín Rayo argues that this is shaped by acceptance or rejection of 'just is'-statements: e.g. 'to be composed of water just is to be composed of H2O'. He offers a novel conception of metaphysical possibility, and a new trivialist philosophy of mathematics.

Reflections on the Foundations of Mathematics

Reflections on the Foundations of Mathematics
Author: Stefania Centrone
Publisher: Springer Nature
Total Pages: 511
Release: 2019-11-11
Genre: Mathematics
ISBN: 3030156559
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This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The volume is divided into three sections, the first two of which focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The third section then builds on this and is centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics.

Reconsidering Logical Positivism

Reconsidering Logical Positivism
Author: Michael Friedman
Publisher: Cambridge University Press
Total Pages: 276
Release: 1999-07-28
Genre: Philosophy
ISBN: 9780521624763
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A reinterpretation of the enduring significance of logical positivism.