The Philosophy of Set Theory by Tiles, Mary -- Read -- Imperial Library of Trantor

Index

Title Page Copyright Page Epigraph Dedication Table of Contents Preface Introduction: Invention or Discovery? 1 - The Finite Universe

1 Finitism 2 Continuity and Infinity 3 Zeno’s Paradoxes 4 The Universe and the Absolutely Infinite

2 - Classes and Aristotelian Logic

1 Aristotelian Logic 2 Aristotle on Knowledge of Universals 3 Nominalism and Extensionalism 4 Toward an Algebra of Classes 5 Classical Finitism in Retreat?

3 - Permutations, Combinations and Infinite Cardinalities

1 Finite Permutations and Combinations 2 Probabilities as Limits of Infinite Sequences 3 Infinite Cardinalities 4 The Finitist Response

4 - Numbering the Continuum

1 The Algebraization of Geometry 2 The Arithmetization of Analysis 3 Toward Infinite Ordinal Numbers 4 Conflict with Classical Finitism

5 - Cantor’s Transfinite Paradise

1 Sets and Cardinal Numbers 2 Transfinite Ordinal Numbers 3 Ordinal Numbers and Cantor’s Continuum Hypothesis 4 Order Types 5 Set Theoretic Paradoxes

6 - Axiomatic Set Theory

1 Axiomatization 2 The ZF Axioms 3 Transfinite Numbers in ZF

7 - Logical Objects and Logical Types

1 Frege, Logic and Arithmetic 2 Frege’s Universe 3 Iterative Sets and Simple Types 4 Russell’s Logicist Reduction

8 - Independence Results and the Universe of Sets

1 Gödel’s Constructible Universe 2 Cardinals and Ordinals in Models 3 Inner Models 4 Generic Sets

9 - Mathematical Structure – Construct and Reality

1 Appeals to Mathematical Intuition 2 Appeals to Mathematical Consequences 3 Descriptive Set Theory 4 Foundations and Superstructures

Further Reading Bibliography Glossary of Symbols Index