Proper and Improper Forcing Perspectives in Logic 2nd Edition by Saharon Shelah ISBN 1107168368 9781107168367

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ISBN 10: 1107168368 
ISBN 13: 9781107168367
Author: Saharon Shelah

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. This volume, the fifth publication in the Perspectives in Logic series, studies set-theoretic independence results (independence from the usual set-theoretic ZFC axioms), in particular for problems on the continuum. The author gives a complete presentation of the theory of proper forcing and its relatives, starting from the beginning and avoiding the metamathematical considerations. No prior knowledge of forcing is required. The book will enable a researcher interested in an independence result of the appropriate kind to have much of the work done for them, thereby allowing them to quote general results.

Proper and Improper Forcing Perspectives in Logic 2nd Table of contents:

Chapter 1: Introduction to Forcing
1.1 Overview of Forcing in Logic
1.2 Historical Development of Forcing
1.3 Key Concepts in Set Theory and Model Theory
1.4 The Role of Forcing in Independence Results
1.5 Applications of Forcing to Mathematical Logic
1.6 Structure of the Book

Chapter 2: Basics of Set Theory and Models
2.1 A Review of Set Theory
2.2 Mathematical Models and Their Role in Forcing
2.3 The Zermelo–Fraenkel Set Theory (ZF) and ZFC
2.4 Axioms and Definitions in Set Theory
2.5 The Axiom of Choice and Its Implications for Forcing

Chapter 3: Introduction to Forcing
3.1 The Formal Definition of Forcing
3.2 Forcing Conditions and Partial Orders
3.3 Generic Sets and Their Properties
3.4 Forcing and the Continuum Hypothesis
3.5 Canonical Forcing Extensions
3.6 Basic Applications in Set Theory

Chapter 4: Proper Forcing
4.1 What Makes a Forcing “Proper”?
4.2 The Role of Proper Forcing in Model Theory
4.3 Applications of Proper Forcing in Set Theory
4.4 Examples of Proper Forcing Extensions
4.5 Properties of Proper Forcing
4.6 The Proper Forcing Axiom and Its Consequences

Chapter 5: Improper Forcing
5.1 Defining Improper Forcing
5.2 Distinguishing Between Proper and Improper Forcing
5.3 The Effects of Improper Forcing on Models
5.4 Common Examples of Improper Forcing
5.5 The Role of Improper Forcing in Proving Independence Results
5.6 Comparison to Other Techniques in Set Theory

Chapter 6: Advanced Forcing Techniques
6.1 Iterated Forcing
6.2 Forcing and Its Use in Proving Consistency Results
6.3 Dense Sets and Their Applications
6.4 Forcing with Large Cardinals
6.5 Extending Models with Forcing

Chapter 7: Forcing in the Context of Logic
7.1 Forcing in First-Order Logic
7.2 Forcing and Modal Logic
7.3 Forcing and Proof Theory
7.4 Logical Consequences of Forcing Extensions
7.5 Forcing and the Philosophy of Logic

Chapter 8: Applications and Implications
8.1 Forcing and the Independence of Axioms
8.2 Forcing and Computability
8.3 Forcing and the Foundation of Mathematics
8.4 Implications for the Continuum Hypothesis and Large Cardinals
8.5 Applications in the Philosophy of Set Theory

Chapter 9: Recent Developments and Research Directions
9.1 New Advances in Forcing Techniques
9.2 Current Research in Proper and Improper Forcing
9.3 The Impact of Forcing on Modern Mathematical Logic
9.4 Future Directions in Set Theory and Model Theory
9.5 Open Problems and Conjectures in Forcing Theory

Chapter 10: Conclusion
10.1 Summary of Key Insights
10.2 The Importance of Forcing in Modern Logic
10.3 Challenges in Understanding Forcing
10.4 The Future of Forcing Theory and Its Applications

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