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Subsystems of Second Order Arithmetic

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ISBN-10: 052188439X

ISBN-13: 9780521884396

Edition: 2nd 2009

Authors: Stephen G. Simpson

List price: $138.95
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Book details

List price: $138.95
Edition: 2nd
Copyright year: 2009
Publisher: Cambridge University Press
Publication date: 5/29/2009
Binding: Hardcover
Pages: 464
Size: 6.50" wide x 9.50" long x 1.25" tall
Weight: 1.738
Language: English

List of Tables
Preface
Acknowledgments
Introduction
The Main Question
Subsystems of Z2
The System ACA0
Mathematics within ACA0
$$-CA0 and Stronger Systems
Mathematics within $$-CA0
The System RCA0
Mathematics within RCA0
Reverse Mathematics
The System WKL0
The System ATR0
The Main Question, Revisited
Outline of Chapters II through X
Conclusions
Development of Mathematics within Subsystems of Z2
Recursive Comprehension
The Formal System RCA0
Finite Sequences
Primitive Recursion
The Number Systems
Complete Separable Metric Spaces
Continuous Functions
More on Complete Separable Metric Spaces
Mathematical Logic
Countable Fields
Separable Banach Spaces
Conclusions
Arithmetical Comprehension
The Formal System ACA0
Sequential Compactness
Strong Algebraic Closure
Countable Vector Spaces
Maximal Ideals in Countable Commutative Rings
Countable Abelian Groups
K&oumlet;nig's Lemma and Ramsey's Theorem
Conclusions
Weak K&oumlet;nig's Lemma
The Heine/Borel Covering Lemma
Properties of Continuous Functions
The G&oumlet;del Completeness Theorem
Formally Real Fields
Uniqueness of Algebraic Closure
Prime Ideals in Countable Commutative Rings
Fixed Point Theorems
Ordinary Differential Equations
The Separable Hahn/Banach Theorem
Conclusions
Arithmetical Transfinite Recursion
Countable Well Orderings; Analytic Sets
The Formal System ATR0
Borel Sets
Perfect Sets; Pseudohierarchies
Reversals
Comparability of Countable Well Orderings
Countable Abelian Groups
�01 and �01 Determinacy
The �01 and �01 Ramsey Theorems
Conclusions
$$ Comprehension
Perfect Kernels
Coanalytic Uniformization
Coanalytic Equivalence Relations
Countable Abelian Groups
�01 � $$ Determinacy
The �02 Ramsey Theorem
Stronger Set Existence Axioms
Conclusions
Models of Subsystems of Z2
�-Models
The Minimum �-Model of $$-CA0
Countable Coded �-Models
A Set-Theoretic Interpretation of ATR0
Constructible Sets and Absoluteness
Strong Comprehension Schemes
Strong Choice Schemes
�-Model Reflection
Conclusions
�-Models
�-Models of RCA0 and ACA0
Countable Coded �-Models of WKL0
Hyperarithmetical Sets
�-Models of �11 Choice
�-Model Reflection and Incompleteness
�-Models of Strong Systems
Conclusions
Non-�-Models
The First Order Parts of RCA0 and ACA0
The First Order Part of WKL0
A Conservation Result for Hilbert's Program
Saturated Models
Gentzen-Style Proof Theory
Conclusions
Additional Results
Measure Theory
Separable Banach Spaces
Countable Combinatorics
Reverse Mathematics for RCA0
Conclusions
Bibliography
Index