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

Name: Philosophy of Set Theory An Historical Introduction to Cantor's Paradise
Availability: InStock
ISBN: 9780486435206

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-…

Book details

List price: $15.95
Copyright year: 2004
Publisher: Dover Publications, Incorporated
Publication date: 6/18/2004
Binding: Paperback
Pages: 256
Size: 5.35" wide x 8.46" long x 0.55" tall
Weight: 0.616
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