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

ISBN-10: 0486435202

ISBN-13: 9780486435206

Edition: 2004

Authors: Mary Tiles
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Description: David Hilbert famously remarked, "No one will drive us from the paradise that Cantor has created." 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; logical objects and logical types; independence results and the universe of sets; and the constructs and reality of mathematical structure. Philosophers and mathematicians will find an abundance of intriguing topics in this text, which is appropriate for undergraduate- and graduate-level courses. 1989 ed. 32 figures.

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Book details

List price: $14.95
Copyright year: 2004
Publisher: Dover Publications, Incorporated
Publication date: 6/18/2004
Binding: Paperback
Pages: 272
Size: 5.25" wide x 8.25" long x 0.50" tall
Weight: 0.594
Language: English

Preface
Introduction: Invention or Discovery?
The Finite Universe
Finitism
Continuity and Infinity
Zeno's Paradoxes
The Universe and the Absolutely Infinite
Classes and Aristotelian Logic
Aristotelian Logic
Aristotle on Knowledge of Universals
Nominalism and Extensionalism
Toward an Algebra of Classes
Classical Finitism in Retreat?
Permutations, Combinations and Infinite Cardinalities
Finite Permutations and Combinations
Probabilities as Limits of Infinite Sequences
Infinite Cardinalities
The Finitist Response
Numbering the Continuum
The Algebraization of Geometry
The Arithmetization of Analysis
Toward Infinite Ordinal Numbers
Conflict with Classical Finitism
Cantor's Transfinite Paradise
Sets and Cardinal Numbers
Transfinite Ordinal Numbers
Ordinal Numbers and Cantor's Continuum Hypothesis
Order Types
Set Theoretic Paradoxes
Axiomatic Set Theory
Axiomatization
The ZF Axioms
Transfinite Numbers in ZF
Logical Objects and Logical Types
Frege, Logic and Arithmetic
Frege's Universe
Iterative Sets and Simple Types
Russell's Logicist Reduction
Independence Results and the Universe of Sets
Godel's Constructible Universe
Cardinals and Ordinals in Models
Inner Models
Generic Sets
Mathematical Structure--Construct and Reality
Appeals to Mathematical Intuition
Appeals to Mathematical Consequences
Descriptive Set Theory
Foundations and Superstructures
Further reading
Bibliography
Glossary of Symbols
Index
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