Reflections on the foundations of mathematics : univalent foundations, set theory and general thoughts / Stefania Centrone, Deborah Kant, Deniz Sarikaya, editors.
This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover sys...
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Full Text (via Springer) |
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| Other Authors: | , , |
| Format: | eBook |
| Language: | English |
| Published: |
Cham :
Springer Nature,
[2019]
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| Series: | Synthese library ;
v. 407. |
| Subjects: |
Table of Contents:
- Intro; Introduction; The Topic; Historical Background; Current Foundations; Set Theory; Homotopy Type Theory/Univalent Foundations; The Contributions; Part I: Current Challenges for the Set-Theoretic Foundations; Part II: What are Homotopy Type Theory and the Univalent Foundations?; Part III: Comparing Set theory, Category Theory, and Type Theory; Part IV: Philosophical Thoughts on the Foundations of Mathematics; Part V: Foundations in Mathematical Practice; The Editors; Literature; Contents; Part I Current Challenges for the Set-Theoretic Foundations; 1 Interview With a Set Theorist.
- Introduction1.1 Introduction; 1.2 Methodological Background; 1.2.1 How to Describe and Analyse Set-Theoretic Practice?; 1.2.2 Why Describe and Analyse Set-Theoretic Practice?; 1.3 Preliminary Facts; 1.4 Some Important Forcing Results; 1.4.1 Cohen's Introduction of Forcing; 1.4.2 Important Forcing Results; 1.4.2.1 Easton Forcing; 1.4.2.2 Suslin's Hypothesis, Iterated Forcing and Martin's Axiom; 1.4.2.3 Laver Forcing; 1.4.2.4 Proper Forcing and Proper Forcing Axiom; 1.5 Philosophical Thoughts in Set Theory; 1.6 Set-Theoretic Intuition About Independence; 1.7 Conclusion; References; References.
- 2 How to Choose New Axioms for Set Theory?2.1 Introduction; 2.2 Ordinary Mathematics; 2.3 Intrinsic Motivations; 2.4 Extrinsic Motivations; 2.5 The Axiom of Constructibility; 2.6 Large Cardinals Axioms; 2.7 Measurable Cardinals and Elementary Embeddings; 2.8 Determinacy Hypotheses; 2.9 Ultimate L and Forcing Axioms; 2.10 Conclusion; References; 3 Maddy On The Multiverse; 3.1 The Problem; 3.2 Multiverse Conceptions; 3.2.1 Naive Multiversism; 3.2.2 Instrumental Multiversism; 3.2.3 Ontological Multiversism; 3.3 Maddy's Assessment of the Multiverse; 3.4 Addressing the Problems.
- 3.4.1 Phenomenology of the Multiverse3.4.1.1 Platonism and Existence; 3.4.1.2 Concepts; 3.4.1.3 Reality and Illusion; 3.4.2 Multiverse-Related Mathematics; 3.4.2.1 Woodin's Set-Generic Multiverse and Ω-logic; 3.4.2.2 The Hyperuniverse Programme; 3.4.2.3 The Multiverse Case for V=L; 3.4.3 Opposing the Argument from Priority and a New Unification; 3.4.4 Relativism Reconsidered; 3.5 Concluding Remarks; Reply to Ternullo on the Multiverse; I. What Is a Multiverse View; II. Naturalistic Concerns About Multiversism; III. Ternullo's Defense; References to Maddy On The Multiverse.