Viale M. The Forcing Method in Set Theory. An Intr.via Boolean Valued Logic 2024

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Viale M. The Forcing Method in Set Theory. An Intr.via Boolean Valued Logic 2024

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Textbook in PDF format The main aim of this book is to provide a compact self-contained presentation of the forcing technique devised by Cohen to establish the independence of the continuum hypothesis from the axioms of set theory. The book follows the approach to the forcing technique via Boolean valued semantics independently introduced by Vopenka and Scott/Solovay; it develops out of notes I prepared for several master courses on this and related topics and aims to provide an alternative (and more compact) account of this topic with respect to the available classical textbooks. The aim of the book is to take up a reader with familiarity with logic and set theory at the level of an undergraduate course on both topics (e.g., familiar with most of the content of introductory books on first-order logic and set theory) and bring her/him to page with the use of the forcing method to produce independence (or undecidability results) in mathematics. Familiarity of the reader with general topology would also be quite helpful; however, the book provides a compact account of all the needed results on this matter. Furthermore, the book is organized in such a way that many of its parts can also be read by scholars with almost no familiarity with first-order logic and/or set theory. The book presents the forcing method outlining, in many situations, the intersections of set theory and logic with other mathematical domains. My hope is that this book can be appreciated by scholars in set theory and by readers with a mindset oriented towards areas of mathematics other than logic and a keen interest in the foundations of mathematics. Preface Acknowledgments Introduction Detailed Content How to Use the Book Some Remarks on the Ontology of Mathematics Preliminaries: Preorders, Topologies, Axiomatizationsof Set Theory Topological Spaces Key Properties of Regular Open Sets Preorders Filters, Antichains, and Predense Sets on Quasi-Orders Axiomatizations of Set Theory The ZFC Axiomatization of Set Theory Boolean Algebras Basic Definitions The Order on Boolean Algebras Boolean Identities Ideals and Morphisms of Boolean Algebras Atomic and Finite Boolean Algebras Examples of Boolean Algebras The Prime Ideal Theorem Stone Spaces of Boolean Algebras Boolean Rings Boolean Algebras as Complemented Distributive Lattices Suprema and Infima of Subsets of a Boolean Algebra Complete Boolean Algebras Complete Boolean Algebras of Regular Open Sets Boolean Completions Some Remarks on Partial Orders and Their Boolean Completions Miscellanea: Completeness, Chain Conditions, and the Measure Algebra The κ-Chain Condition The Algebra of Lebesgue Measurable Sets Modulo Null Sets More on Preorders Generic Filters The Quasi-Orders Fn(X,Y) The Quasi-Order Fn(ω2ω,2) Quasi-Orders with the Countable Chain Condition and the -System Lemma Boolean Valued Models Boolean Valued Models and Boolean Valued Semantics Soundness and Completeness for Boolean Valued Semantics Boolean Morphisms A Discussion on Boolean Valued Semantics Quotients of Boolean Valued Models, Fullness, Łoś Theorem Examples of Quotients Counterexamples Łoś Theorem for Full Boolean Valued Models Forcing and Fullness The Mixing Property and Fullness Examples of Boolean Valued Models with the Mixing Property Example I: Spaces of Measurable Functions Example II: Standard Ultraproducts Example III: C(St(B),2ω) Forcing Boolean Valued Models for Set Theory External Definition of Forcing M-Generic Ultrafilters, and the Induced Valuation Map How to Describe an M-generic Filter G for 2