Format:
1 Online-Ressource (277 pages)
ISBN:
9783319629353
Content:
Intro -- Preface -- Introduction: On the Development of the Hyperuniverse Project -- The Philosophical Grounding of the HP -- The Mathematical Development of the HP -- The Chapters in Brief -- Contents -- Class Forcing in Class Theory -- 1 Introduction -- 2 Morse-Kelley Class Theory -- 3 Generics, Names and the Extension -- 4 Definability and Truth Lemmas -- 5 The Extension Fulfills the Axioms -- 6 Laver's Theorem -- References -- Hyperclass Forcing in Morse-Kelley Class Theory -- 1 Introduction -- 2 Coding Between MK* and SetMK* -- 3 Hyperclass Forcing and Forcing in SetMK** -- 4 Minimal β-Models of MK** -- 5 Further Work and Open Questions -- References -- Multiverse Conceptions in Set Theory -- 1 The Set-Theoretic Multiverse -- 1.1 Introduction -- 1.2 The `Universe View' and the `Multiverse View' -- 1.3 A Proposed Classification -- 2 Multiverse Conceptions -- 2.1 The Realist Multiverse View -- 2.1.1 The General Framework -- 2.1.2 Problems with the Realist Multiverse View -- 2.2 The Non-realist Multiverse View -- 2.3 The Set-Generic Multiverse -- 2.3.1 Woodin -- 2.3.2 Steel's Programme -- 2.3.3 Problems with the Set-Generic Multiverse -- 3 The `Vertical' Multiverse -- 3.1 Actualism and Potentialism -- 3.2 Zermelo's Account: the `Vertical' Multiverse -- 3.3 The `Vertical' Multiverse Within the Hyperuniverse Programme -- 3.3.1 A Brief Review of the Hyperuniverse Programme -- 3.3.2 Width Actualism and Infinitary Logic -- 4 Concluding Summary -- References -- Evidence for Set-Theoretic Truth and the Hyperuniverse Programme -- 1 Introduction -- 2 Set-Theoretic Practice -- 3 Set Theory as a Foundation for Mathematics -- 4 The Maximality of the Set-Theoretic Universe and the HP -- 4.1 The Iterative Conception of Set -- 4.2 Maximality and the Iterative Conception -- 4.3 Actualism and Potentialism -- 4.4 Maximality in Height and #-Generation
Content:
4.5 Maximality in Width and the IMH -- 4.6 The Reduction to the Hyperuniverse -- 4.7 Synthesis -- 4.8 The Strong IMH -- 4.9 A Maximality Protocol -- 4.10 CardMax(κ+) (for κ an Infinite Cardinal) -- 4.11 Width Indiscernibility -- 4.12 Omniscience -- 4.13 The Future of the HP -- References -- On the Set-Generic Multiverse -- 1 The Category of Forcing Extensions as the Set-Theoretic Multiverse -- 2 Laver's Theorem and Bukovský's Theorem -- 3 A Formal Deductive System for L∞(μ) -- 4 An Axiomatic Framework for the Set-Generic Multiverse -- 5 Independent Buttons -- References -- On Strong Forms of Reflection in Set Theory -- 1 Introduction -- 2 Reflection with Elementary Embeddings -- 2.1 Embeddings Internal to V -- 2.2 Sharp-Generated Reflection -- 3 An Application -- 3.1 Vertically Maximal Models and IMH -- 3.2 IMH# is Compatible with Large Cardinals -- References -- Definability of Satisfaction in Outer Models -- 1 Introduction -- 2 Preliminaries -- 2.1 A Theorem of Barwise and the Notion of an Outer Model -- 2.2 The Outer Model Theory -- 2.3 Notational Conventions -- 3 A Simplified Case -- 4 Main Result -- 4.1 Good Iterations -- 4.2 Compositions of Good Iterations -- 4.3 Main Theorem -- 4.3.1 Statement and Motivation -- 4.3.2 Proof -- 4.4 Some Generalizations -- 5 Open Questions -- Appendix -- Omniscience from Large Cardinals -- Forcing Omniscience -- References -- The Search for New Axioms in the Hyperuniverse Programme -- 1 New Set-Theoretic Axioms -- 2 Intrinsic Evidence for New Axioms -- 2.1 Brief Remarks on Ontology and Truth -- 2.2 Two Sources of Evidence -- 2.3 The Maximum Iterative Concept -- 3 Conceptions of V: The `Vertical' Multiverse -- 3.1 What is V? The Actualism/Potentialism Dichotomy -- 3.2 Zermelo's Account: A `Vertical' Multiverse -- 4 The Hyperuniverse (H): V-Logic -- 4.1 The Hyperuniverse -- 4.2 V-Logic
Content:
5 Maximality Principles for V -- 6 New Axioms as H-Axioms -- 6.1 The Nature of H-Axioms -- 6.2 Alternative Approaches -- 7 Concluding Summary -- Appendix: Absoluteness Axioms -- References -- Explaining Maximality Through the Hyperuniverse Programme -- 1 The Maximal Iterative Concept of Set -- 1.1 Generalities -- 1.2 Expanding on the Maximal Concept -- 1.3 New Intrinsically Justified Axioms -- 2 A Zermelian Approach to V -- 2.1 Height and Width Maximality -- 2.2 Actualism and Potentialism: Zermelo's Conception -- 3 Height Maximality: Reflection -- 4 Width Maximality: V-Logic, IMH -- 4.1 The Strategy -- 4.2 V-Logic and IMH -- 5 Reduction to H -- 5.1 Reduction of IMH -- 5.2 Reduction of #-Generated V: #-Generation Revisited -- 6 H-Axioms: Synthesis of #-Generation with IMH-Variants -- 7 The Dynamic Search for Truth -- References -- Large Cardinals and the Continuum Hypothesis -- 1 Introduction -- 2 How to Find Large Cardinals -- 2.1 Inaccessible Cardinals -- 2.2 Mahlo Cardinals -- 2.3 Analogies with ω -- 2.4 Compact, Measurable, and Ramsey Cardinals -- 2.5 Motivation -- 3 Large Cardinals and CH -- 3.1 How to Force CH and CH -- 3.2 Inaccessible and Mahlo Cardinals and CH -- 3.3 Weakly Compact and Measurable Cardinals and CH -- 3.4 A Uniform Approach -- 3.5 Other Large Cardinals -- 4 On the Consistency Strength -- 5 Conclusion -- References -- Gödel's Cantorianism -- 1 Introductory Remarks -- 2 Intra-Subjective (``Immanent'') Existence -- 3 Concepts as Objective Constructs -- 4 Anti-subjectivism -- 5 Set-Theoretic Platonism -- 6 The Absolute Infinite and the Universe of Sets -- 7 Trans-Subjective (``Transient'') Existence -- 8 Connection of Immanent and Transient -- 9 The Development of Mathematics -- 10 Concluding Remarks -- References -- Remarks on Buzaglo's Concept Expansion and Cantor's Transfinite -- 1 Preliminaries -- 2 Concept Forcing
Content:
2.1 Preliminaries -- 2.2 Formalisation of Concept Expansion -- 3 Cantor on Concept Expansion -- 4 The Creation of the Transfinite -- 5 Concluding Remarks -- Reference
Additional Edition:
9783319629346
Additional Edition:
Print version Antos, Carolin The Hyperuniverse Project and Maximality Cham : Birkhauser Verlag GmbH,c2018 9783319629346
Language:
English
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