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The Resource Proof theory : the first step into impredicativity, Wolfram Pohlers

Proof theory : the first step into impredicativity, Wolfram Pohlers

Creator
Author
Summary
This book verifies with compelling evidence the author's intent to "write a book on proof theory that needs no previous knowledge of proof theory". Avoiding the cryptic terminology of proof theory as far as possible, the book starts at an elementary level and displays the connections between infinitary proof theory and generalized recursion theory, especially the theory of inductive definitions. As a "warm up" Gentzen's classical analysis of pure number theory is presented in a more modern terminology, followed by an explanation and proof of the famous result of Feferman and Schütte on the limits of predicativity. The author also provides an introduction to ordinal arithmetic, introduces the Veblen hierarchy and employs these functions to design an ordinal notation system for the ordinals below Epsilon 0 and Gamma 0, while emphasizing the first step into impredicativity, that is, the first step beyond Gamma 0. This is first done by an analysis of the theory of non-iterated inductive definitions using Buchholz's improvement of local predicativity, followed by Weiermann's observation that Buchholz's method can also be used for predicative theories to characterize their provably recursive functions. A second example presents an ordinal analysis of the theory of $/Pi_2$ reflection, a subsystem of set theory that is proof-theoretically equivalent to Kripke-Platek set. The book is pitched at undergraduate/graduate level, and thus addressed to students of mathematical logic interested in the basics of proof theory. It can be used for introductory as well as more advanced courses in proof theory. An earlier version of this book was published in 1989 as volume 1407 of the "Lecture Notes in Mathematics" (ISBN 978-3-540-51842-6)
Language
eng
Work
Publication
Extent
1 online resource (xiii, 370 pages).
Contents
  • 1 Historical Background
  • 2 Primitive Recursive Functions and Relations
  • 3 Ordinals
  • 4 Pure Logic
  • 5 Truth Complexities for Pi 1-1-Sentences
  • 6 Inductive Definitions
  • 7 The Ordinal Analysis for Pean Arithmetic
  • 8 Autonomous Ordinals and the Limits of Predicativity
  • 9 Ordinal Analysis of the Theory for Inductive Definitions
  • 10 Provably Recursive Functions of NT
  • 11 Ordinal Analysis for Kripke Platek Set Theory with infinity
  • 12 Predicativity Revisited
  • 13 Non-Monotone Inductive Definitions
  • 14 Epilogue
Isbn
9783540693192
Instance
Label
Proof theory : the first step into impredicativity
Title
Proof theory
Title remainder
the first step into impredicativity
Statement of responsibility
Wolfram Pohlers
Creator
Author
Subject
Language
eng
Summary
This book verifies with compelling evidence the author's intent to "write a book on proof theory that needs no previous knowledge of proof theory". Avoiding the cryptic terminology of proof theory as far as possible, the book starts at an elementary level and displays the connections between infinitary proof theory and generalized recursion theory, especially the theory of inductive definitions. As a "warm up" Gentzen's classical analysis of pure number theory is presented in a more modern terminology, followed by an explanation and proof of the famous result of Feferman and Schütte on the limits of predicativity. The author also provides an introduction to ordinal arithmetic, introduces the Veblen hierarchy and employs these functions to design an ordinal notation system for the ordinals below Epsilon 0 and Gamma 0, while emphasizing the first step into impredicativity, that is, the first step beyond Gamma 0. This is first done by an analysis of the theory of non-iterated inductive definitions using Buchholz's improvement of local predicativity, followed by Weiermann's observation that Buchholz's method can also be used for predicative theories to characterize their provably recursive functions. A second example presents an ordinal analysis of the theory of $/Pi_2$ reflection, a subsystem of set theory that is proof-theoretically equivalent to Kripke-Platek set. The book is pitched at undergraduate/graduate level, and thus addressed to students of mathematical logic interested in the basics of proof theory. It can be used for introductory as well as more advanced courses in proof theory. An earlier version of this book was published in 1989 as volume 1407 of the "Lecture Notes in Mathematics" (ISBN 978-3-540-51842-6)
Member of
Cataloging source
MND
http://library.link/vocab/creatorName
Pohlers, Wolfram
Dewey number
511.36
Image bit depth
0
Index
index present
Language note
English
LC call number
QA9.54
LC item number
.P645 2009
Literary form
non fiction
Nature of contents
  • dictionaries
  • bibliography
Series statement
Universitext
http://library.link/vocab/subjectName
  • Proof theory
  • Logic, Symbolic and mathematical
  • Logic, Symbolic and mathematical
  • Beweistheorie
Label
Proof theory : the first step into impredicativity, Wolfram Pohlers
Instantiates
Publication
Antecedent source
mixed
Bibliography note
Includes bibliographical references (pages 357-361) and index
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
1 Historical Background -- 2 Primitive Recursive Functions and Relations -- 3 Ordinals -- 4 Pure Logic -- 5 Truth Complexities for Pi 1-1-Sentences -- 6 Inductive Definitions -- 7 The Ordinal Analysis for Pean Arithmetic -- 8 Autonomous Ordinals and the Limits of Predicativity -- 9 Ordinal Analysis of the Theory for Inductive Definitions -- 10 Provably Recursive Functions of NT -- 11 Ordinal Analysis for Kripke Platek Set Theory with infinity -- 12 Predicativity Revisited -- 13 Non-Monotone Inductive Definitions -- 14 Epilogue
Control code
311303031
Dimensions
unknown
Extent
1 online resource (xiii, 370 pages).
File format
multiple file formats
Form of item
online
Isbn
9783540693192
Level of compression
uncompressed
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other control number
10.1007/978-3-540-69319-2
http://library.link/vocab/ext/overdrive/overdriveId
978-3-540-69318-5
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
(OCoLC)311303031
Label
Proof theory : the first step into impredicativity, Wolfram Pohlers
Publication
Antecedent source
mixed
Bibliography note
Includes bibliographical references (pages 357-361) and index
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
1 Historical Background -- 2 Primitive Recursive Functions and Relations -- 3 Ordinals -- 4 Pure Logic -- 5 Truth Complexities for Pi 1-1-Sentences -- 6 Inductive Definitions -- 7 The Ordinal Analysis for Pean Arithmetic -- 8 Autonomous Ordinals and the Limits of Predicativity -- 9 Ordinal Analysis of the Theory for Inductive Definitions -- 10 Provably Recursive Functions of NT -- 11 Ordinal Analysis for Kripke Platek Set Theory with infinity -- 12 Predicativity Revisited -- 13 Non-Monotone Inductive Definitions -- 14 Epilogue
Control code
311303031
Dimensions
unknown
Extent
1 online resource (xiii, 370 pages).
File format
multiple file formats
Form of item
online
Isbn
9783540693192
Level of compression
uncompressed
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other control number
10.1007/978-3-540-69319-2
http://library.link/vocab/ext/overdrive/overdriveId
978-3-540-69318-5
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
(OCoLC)311303031

Subject

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