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E-raamat: Recursion-Theoretic Hierarchies

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(University of Michigan, Ann Arbor)
  • Formaat: PDF+DRM
  • Sari: Perspectives in Logic
  • Ilmumisaeg: 02-Mar-2017
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316731666
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Recursion-Theoretic HierarchiesSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Perspectives in Logic
  • Ilmumisaeg: 02-Mar-2017
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316731666
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Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. The theory set out in this volume, the ninth publication in the Perspectives in Logic series, is the result of the meeting and common development of two currents of mathematical research: descriptive set theory and recursion theory. Both are concerned with notions of definability and with the classification of mathematical objects according to their complexity. These are the common themes which run through the topics discussed here. The author develops a general theory from which the results of both areas can be derived, making these common threads clear.

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The theory set out in this book results from the meeting of descriptive set theory and recursion theory.
Introduction 1(4)
Part A Basic Notions of Definability
5(128)
Chapter I Groundwork
7(20)
1 Logic and Set Theory
7(8)
2 Topology and Measure
15(7)
3 Inductive Definitions
22(5)
Chapter II Ordinary Recursion Theory
27(42)
1 Primitive Recursion
28(9)
2 Recursive Functionals and Relations
37(9)
3 Normal Forms
46(6)
4 Semi-Recursive Relations
52(10)
5 Relativization
62(7)
Chapter III Hierarchies and Definability
69(64)
1 The Arithmetical Hierarchy
69(11)
2 The Analytical Hierarchy
80(9)
3 Inductive Definability
89(17)
4 Implicit Definability and Bases
106(8)
5 Definability in Formal Languages for Arithmetic
114(10)
6 Arithmetical Forcing
124(9)
Part B The Analytical and Projective Hierarchies
133(124)
Chapter IV The First Level
135(66)
1 Π11 and Well-Orderings
135(8)
2 The Boundedness Principle and Other Applications
143(13)
3 The Borel Hierarchy
156(7)
4 The Effective Borel and Hyperarithmetical Hierarchies
163(17)
5 Cardinality, Measurability and Category
180(8)
6 Continuous Images
188(6)
7 Uniformization
194(7)
Chapter V Δ12 and Beyond
201(56)
1 The Pre-Wellordering Property
202(12)
2 The Hypothesis of Constructibility
214(7)
3 The Hypothesis of Projective Determinacy
221(15)
4 Classical Hierarchies in Δ1r
236(10)
5 Effective Hierarchies in Δ1r
246(5)
6 A Hierarchy for Δ12
251(6)
Part C Generalized Recursion Theories
257(188)
Chapter VI Recursion in a Type-2 Functional
259(84)
1 Basic Properties
259(12)
2 Substitution Theorems
271(13)
3 Ordinal Comparison
284(7)
4 Relations Semi-Recursive in a Type-2 Functional
291(16)
5 Hierarchies of Relations Recursive in a Type-2 Functional
307(8)
6 Extended Functionals
315(20)
7 Recursive Type-3 Functionals and Relations
335(8)
Chapter VII Recursion in a Type-3 Functional
343(28)
1 Basic Properties
343(7)
2 Relations Semi-Recursive in a Type-3 Functional
350(10)
3 Hierarchies of Relations Recursive in a Type-3 Functional
360(3)
4 Higher Types
363(8)
Chapter VIII Recursion on Ordinals
371(74)
1 Recursive Ordinal Functions
372(11)
2 Recursively Regular Ordinals
383(10)
3 Ordinal Recursion and the Analytical Hierarchy
393(10)
4 Ordinal Recursion and Type-2 Functionals
403(9)
5 Stability
412(7)
6 Recursively Large Ordinals
419(11)
7 Ordinal Recursion and Constructible Sets
430(15)
Epilogue 445(14)
References 459(8)
Global Notational Conventions 467(2)
Special Notations 469(4)
Index 473
Peter G. Hinman works in the Department of Mathematics at the University of Michigan, Ann Arbor.
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