Viale M. The Forcing Method in Set Theory. An Intr.via Boolean Valued Logic 2024
Download this torrent!
Viale M. The Forcing Method in Set Theory. An Intr.via Boolean Valued Logic 2024
To start this P2P download, you have to install a BitTorrent client like qBittorrent
Category: Other
Total size: 5.55 MB
Added: 2025-03-10 23:39:04
Share ratio:
9 seeders,
4 leechers
Info Hash: D3126B26950B89341F8DC82E78DA77D311F376C8
Last updated: 13.5 hours ago
Description:
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