A Logical Foundation for Potentialist Set Theory

A new approach to the standard axioms of set theory, relating the theory to the philosophy of science and metametaphysics.

Sharon Berry (Author)

9781108834315, Cambridge University Press

Hardback, published 17 February 2022

288 pages
25.1 x 17.3 x 1.9 cm, 0.598 kg

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.

1. Introduction
Part I: 2. Actualist set theory
3. Putnamian potentialism: Putnam and Hellman
4. Overview of my proposal
5. Parsonian potentialism
Part II: 6. Purified potentialist set theory: An informal sketch
7. Content restriction
8. Inference rules
9. Defense of ZFC
Part III: 10. Platonism or nominalism?
11. Indispensability
12. Modal If-thenist paraphrase strategy
13. Explanatory indispensability
14. Physical magnitude statements and sparsity
15. Weak quantifier variance and mathematical objects
16. Weak quantifier variance, knowledge by stipulative definition and access worries
17. Logicism and structuralism
18. Anti-objectivism about set theory
19. Conclusion
Bibliography
Index.

Subject Areas: Philosophy of science [PDA], Philosophy of mathematics [PBB], Philosophy: logic [HPL]