The Philosophy of Set Theory An Historical Introduction to Cantor's Paradise

by Tiles, Mary

• ISBN13: 9780486435206
• ISBN10: 0486435202
• Format: Paperback
• Copyright: 2004-06-18
• Publisher: Dover Publications

Summary

This volume offers a guided tour of modern mathematics' Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor's transfinite paradise; axiomatic set theory, and more. 1989 edition. Includes 32 figures.

Table of Contents

Preface	xiii
Introduction: Invention or Discovery?	1	(5)
1 The Finite Universe	6	(26)
1 Finitism	6	(4)
2 Continuity and Infinity	10	(2)
3 Zeno's Paradoxes	12	(10)
4 The Universe and the Absolutely Infinite	22	(10)
2 Classes and Aristotelian Logic	32	(24)
1 Aristotelian Logic	33	(6)
2 Aristotle on Knowledge of Universals	39	(8)
3 Nominalism and Extensionalism	47	(3)
4 Toward an Algebra of Classes	50	(4)
5 Classical Finitism in Retreat?	54	(2)
3 Permutations, Combinations and Infinite Cardinalities	56	(12)
1 Finite Permutations and Combinations	57	(2)
2 Probabilities as Limits of Infinite Sequences	59	(3)
3 Infinite Cardinalities	62	(2)
4 The Finitist Response	64	(4)
4 Numbering the Continuum	68	(27)
1 The Algebraization of Geometry	69	(15)
2 The Arithmetization of Analysis	84	(4)
3 Toward Infinite Ordinal Numbers	88	(2)
4 Conflict with Classical Finitism	90	(5)
5 Cantor's Transfinite Paradise	95	(23)
1 Sets and Cardinal Numbers	96	(8)
2 Transfinite Ordinal Numbers	104	(3)
3 Ordinal Numbers and Cantor's Continuum Hypothesis	107	(4)
4 Order Types	111	(3)
5 Set Theoretic Paradoxes	114	(4)
6 Axiomatic Set Theory	118	(20)
1 Axiomatization	118	(3)
2 The ZF Axioms	121	(13)
3 Transfinite Numbers in ZF	134	(4)
7 Logical Objects and Logical Types	138	(37)
1 Frege, Logic and Arithmetic	138	(8)
2 Frege's Universe	146	(8)
3 Iterative Sets and Simple Types	154	(4)
4 Russell's Logicist Reduction	158	(17)
8 Independence Results and the Universe of Sets	175	(17)
1 Godel's Constructible Universe	175	(4)
2 Cardinals and Ordinals in Models	179	(4)
3 Inner Models	183	(2)
4 Generic Sets	185	(7)
9 Mathematical Structure - Construct and Reality	192	(32)
1 Appeals to Mathematical Intuition	192	(6)
2 Appeals to Mathematical Consequences	198	(3)
3 Descriptive Set Theory	201	(6)
4 Foundations and Superstructures	207	(17)
Further reading	224	(3)
Bibliography	227	(6)
Glossary of Symbols	233	(2)
Index	235