Ordinal proof theory of Kripke-Platek set theory augmented by strong reflection principles [Elektronische Ressource] / Jan-Carl Stegert. Betreuer: Wolfram Pohlers

Ordinal proof theory of Kripke-Platek set theory augmented by strong reflection principles [Elektronische Ressource] / Jan-Carl Stegert. Betreuer: Wolfram Pohlers

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Jan-Carl StegertOrdinal Proof Theory of Kripke-PlatekSet Theory Augmented by StrongRe ection Principles-2010-Mathematische LogikOrdinal Proof Theory of Kripke-PlatekSet Theory Augmented by StrongRe ection PrinciplesInaugural-Dissertationzur Erlangung des akademischen Gradeseines Doktors der Naturwissenschaftendurch den Fachbereich Mathematik und Informatikder Westfalischen Wilhelms-Universitat Munster vorgelegt vonJan-Carl Stegertaus Witten-2010-Dekan: Prof. Dr. Matthias Low eErster Gutachter: Prof. Dr. Wolfram PohlersZweiter Gutachter: Prof. Dr. Wilfried BuchholzTag der mundlichen Prufung: 30.03.2011 Tag der Promotion:ContentsPreface 3I. An Ordinal Analysis of -Ref 7!1. Introduction 82. Ordinal Theory 1312.1. -Indescribable Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . 13n2.2. Collapsing Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3. Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4. The Ordinal Notation System T( ) . . . . . . . . . . . . . . . . . . . . . 293. The Fine Structure of the Collapsing Hierarchies 313.1. Path Fidelity and Correctness . . . . . . . . . . . . . . . . . . . . . . . . 313.2. The Domination Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 364. A semi-formal Calculus for -Ref 41!4.1. Rami ed Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2. Semi-formal Derivations on Hull-Sets of T( ) .

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Jan-Carl Stegert
Ordinal Proof Theory of Kripke-Platek
Set Theory Augmented by Strong
Re ection Principles
-2010-Mathematische Logik
Ordinal Proof Theory of Kripke-Platek
Set Theory Augmented by Strong
Re ection Principles
Inaugural-Dissertation
zur Erlangung des akademischen Grades
eines Doktors der Naturwissenschaften
durch den Fachbereich Mathematik und Informatik
der Westfalischen Wilhelms-Universitat Munster
vorgelegt von
Jan-Carl Stegert
aus Witten
-2010-Dekan: Prof. Dr. Matthias Low e
Erster Gutachter: Prof. Dr. Wolfram Pohlers
Zweiter Gutachter: Prof. Dr. Wilfried Buchholz
Tag der mundlichen Prufung: 30.03.2011
Tag der Promotion:Contents
Preface 3
I. An Ordinal Analysis of -Ref 7!
1. Introduction 8
2. Ordinal Theory 13
12.1. -Indescribable Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . 13n
2.2. Collapsing Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3. Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4. The Ordinal Notation System T( ) . . . . . . . . . . . . . . . . . . . . . 29
3. The Fine Structure of the Collapsing Hierarchies 31
3.1. Path Fidelity and Correctness . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2. The Domination Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4. A semi-formal Calculus for -Ref 41!
4.1. Rami ed Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2. Semi-formal Derivations on Hull-Sets of T( ) . . . . . . . . . . . . . . . 43
4.3. Embedding of -Ref . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47!
5. Cut and Re ection Elimination Theorems 51
5.1. Predicative Cut Elimination . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2. Re ection Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Review: Ordinal Analysis of -Ref 63n
Singular Collapsing: -Ref . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
Simultaneous Collapsing: -Ref . . . . . . . . . . . . . . . . . . . . . . . . . 653
Iterated Simultaneous Collapsing: -Ref . . . . . . . . . . . . . . . . . . . . 68n
II. The Provable Recursive Functions of -Ref 71!
6. Introduction 72
17. Subrecursive Hierarchies 74

7.1. The Theory -Ref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74!
7.2. The Finite Content of Ordinals, Terms and Sentences . . . . . . . . . . 75
7.3. Subrecursive Hierarchies on Hull-Sets of T( ) . . . . . . . . . . . . . . . 75

8. A re ned semi-formal Calculus for -Ref 77!
8.1. Semi-formal Derivations on Fragmented Hull-Sets of T( ) . . . . . . . . 77

8.2. Embedding of -Ref . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79!
9. Cut and Re ection Elimination Theorems for Re ned Derivations 86
9.1. Predicative Cut Elimination for In nitary Derivations on Fragmented
Hull-Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
9.2. Re ection Elimination for In nitary Derivations on Fragmented Hull-Sets 88
10.A Characterization of the Provable Recursive Functions of -Ref 91!
10.1. Collapsing and the Witnessing Theorem . . . . . . . . . . . . . . . . . . 91
10.2. A Subrecursive Hierarchy Dominating the Provable Recursive Functions
of -Ref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94!
III. An Ordinal Analysis of Stability 97
11.Introduction 98
12.Ordinal Theory for Stability 100
12.1.-Indescribable Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . 100
12.2. Collapsing Hierarchies based on -Indescribable Cardinals . . . . . . . . 101
12.3. Structure Theory of Stability . . . . . . . . . . . . . . . . . . . . . . . . . 109
12.4. The Ordinal Notation System T( ) . . . . . . . . . . . . . . . . . . . . . 113
13.The Fine Structure Theory 114
13.1. The Domination Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 117
14.A semi-formal Calculus for Stability 122
~14.1. The Compound LanguageL . . . . . . . . . . . . . . . . . . . . . . 122RS( )
14.2. Semi-formal Derivations on Hull-Sets of T( ) . . . . . . . . . . . . . . . 124
14.3. Embedding of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
15.Elimination of Re ection Rules 131
~15.1. Useful Properties ofL . . . . . . . . . . . . . . . . . . . . . . . . . 131
RS( )
15.2. Re ection Elimination for Stability . . . . . . . . . . . . . . . . . . . . . 133
IV. The Provable Recursive Functions of Stability 143
16.A Characterization of the Provable Recursive Functions of Stability 144
2Preface
This thesis belongs to the area of mathematical logic, more precisely to the area of
proof theory. One of the topics of proof theory arose from the famous talk given by
David Hilbert in 1900 at Paris. Shattered by the paradoxes of set theory discovered
1at that time he suggested a research programme, based on two pillars: At rst he
demanded an axiomatization of the existing mathematics and at second he proposed
the establishment of metamathematics (proof theory) to secure the consistency of this
axiomatization by pure