Understanding the Infinite

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Harvard University Press, 1994 - Mathematics - 372 pages
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How can the infinite, a subject so remote from our finite experience, be an everyday working tool for the working mathematician? Blending history, philosophy, mathematics and logic, Shaughan Lavine answers this question with clarity. An account of the origins of the modern mathematical theory of the infinite, his book is also a defense against the attacks and misconceptions that have dogged this theory since its introduction in the late 19th century.
 

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User Review  - fpagan - LibraryThing

Deep discussion of the mathematical work of Cantor et al, and proposal for "a finite mathematics of indefinitely large size." Read full review

Contents

Introduction
1
II Infinity Mathematics Persistent Suitor
11
1 Incommensurable Lengths Irrational Numbers
12
2 Newton and Leibniz
15
3 Go Forward and Faith Will Come to You
22
4 Vibrating Strings
26
5 Infinity Spurned
32
6 Infinity Embraced
37
Knowing the Infinite 1 What Do We Know?
154
2 What Can We Know?
162
3 Getting from Here to There
181
4 Appendix
203
Leaps of Faith 1 Intuition
213
2 Physics
218
3 Modality
221
4 SecondOrder Logic
224

Sets of Points 1 Infinite Sizes
42
2 Infinite Orders
44
3 Integration
49
4 Absolute vs Transfinite
51
5 Paradoxes
57
What Are Sets? 1 Russell
63
2 Cantor
76
Letter from Cantor to Jourdain 9 July 1904
98
On an Elementary Question of Set Theory
99
1 The Axiom of Choice
103
2 The Axiom of Replacement
119
3 Definiteness and Skolems Paradox
123
4 Zermelo
134
5 Go Forward and Faith Will Come to You
141
1 Who Needs SelfEvidence?
241
2 Picturing the Infinite
246
3 The Finite Mathematics of Indefinitely Large Size
267
4 The Theory of Zillions
288
Extrapolations 1 Natural Models
309
2 Many Models
314
3 One Model or Many? Sets and Classes
316
4 Natural Axioms
320
5 Second Thoughts
322
6 Schematic and Generalizable Variables
325
Bibliography
329
Index
349
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About the author (1994)

Shaughan Lavine is Associate Professor of Philosophy at the University of Arizona.

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