The Evolution of LogicExamines the relations between logic and philosophy over the last 150 years. Logic underwent a major renaissance beginning in the nineteenth century. Cantor almost tamed the infinite, and Frege aimed to undercut Kant by reducing mathematics to logic. These achievements were threatened by the paradoxes, like Russell's. This ferment generated excellent philosophy (and mathematics) by excellent philosophers (and mathematicians) up to World War II. This book provides a selective, critical history of the collaboration between logic and philosophy during this period. After World War II, mathematical logic became a recognized subdiscipline in mathematics departments, and consequently but unfortunately philosophers have lost touch with its monuments. This book aims to make four of them (consistency and independence of the continuum hypothesis, Post's problem, and Morley's theorem) more accessible to philosophers, making available the tools necessary for modern scholars of philosophy to renew a productive dialogue between logic and philosophy. |
Contents
| 1 | |
Die Urwahrheiten | 31 |
Which Sets Exist? | 59 |
The Universe and Everything | 89 |
Truth Eludes Proof | 123 |
Accommodating Cantor | 152 |
Or Not | 175 |
The Critique of Pure Reason | 205 |
The Ways of the World | 235 |
The Zoology of Reality | 268 |
| 281 | |
| 291 | |
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Common terms and phrases
abstract algorithm analytic argument axiom of choice axiomatic binary called Cantor’s cardinal Church–Turing thesis computable condition constant constructible sets continuum hypothesis converges countable define denoted domain elementary extension elementary submodel extensionality first-order fodo formula Frege function whose value gödel number Hence induction infinity initial segment integers isomorphic Kant Kurt Gödel language of ZF limited sentence maps model for ZF Model Theory N-language n-tuples natural numbers negation nonempty one–one ordered pairs ordinals paradox partial algorithm partial recursive philosophy polyadicity power set predicate primitive priori proof proposition provable prove quantifiers recursive function relation replacement Russell Russell’s says second-order semantic set of ordered set theory singular terms Skolem Skolem hull strongly minimal Suppose Tarski theorem tion true truth truth-functional unary uncountable undecidable union variables Vaught pair W. V. Quine weakly forces well-order xa I F(x Zermelo