The axiom of determinacy forcing axioms and the nonstationary ideal 1st Edition by W Hugh Woodin ISBN 3110197022 9783110197020 by Woodin W.h. 9783110197020, 3110197022 instant download after payment.

The axiom of determinacy forcing axioms and the nonstationary ideal 1st Edition by W Hugh Woodin - Ebook PDF Instant Download/Delivery: 3110197022, 9783110197020
Full download The axiom of determinacy forcing axioms and the nonstationary ideal 1st Edition after payment

Product details:

ISBN 10: 3110197022 
ISBN 13: 9783110197020
Author: W Hugh Woodin

The starting point for this monograph is the previously unknown connection between the Continuum Hypothesis and the saturation of the non-stationary ideal on ω₁; and the principle result of this monograph is the identification of a canonical model in which the Continuum Hypothesis is false. This is the first example of such a model and moreover the model can be characterized in terms of maximality principles concerning the universal-existential theory of all sets of countable ordinals. This model is arguably the long sought goal of the study of forcing axioms and iterated forcing but is obtained by completely different methods, for example no theory of iterated forcing whatsoever is required. The construction of the model reveals a powerful technique for obtaining independence results regarding the combinatorics of the continuum, yielding a number of results which have yet to be obtained by any other method.

This monograph is directed to researchers and advanced graduate students in Set Theory. The second edition is updated to take into account some of the developments in the decade since the first edition appeared, this includes a revised discussion of Ω-logic and related matters.

The axiom of determinacy forcing axioms and the nonstationary ideal 1st Table of contents:

Part I: Foundational Concepts and Preliminaries

• Chapter 1: Review of Zermelo-Fraenkel Set Theory (ZFC)
  • Axioms of ZFC
  • Ordinals and Cardinals: Basic Properties
  • Constructible Universe L and Its Properties
  • Inner Models and Relative Consistency
• Chapter 2: Filters, Ideals, and Combinatorics on Cardinals
  • Definitions of Filters and Ideals
  • The Club and Stationary Filter/Ideal on ω1​
  • Flipping the Ideal and the Pressing Down Lemma
  • Combinatorial Principles on Cardinals (e.g., ♢, □)

Part II: The Axiom of Determinacy (AD)

• Chapter 3: Introduction to Determinacy and Games
  • Infinite Games on the Naturals (ωω)
  • Determinacy of Games: Winning Strategies
  • The Axiom of Determinacy (AD) and Its Initial Implications
  • Projective Sets and Their Regularity Properties under AD
• Chapter 4: Consequences of AD for the Real Line
  • Measurability of Projective Sets
  • The Baire Property and Perfect Set Property for Projective Sets
  • Relationship to Lebesgue Measure and Continuum Hypothesis
• Chapter 5: AD and Large Cardinals
  • Connection of Determinacy to Woodin Cardinals and Beyond
  • Inner Models for AD and Large Cardinals (e.g., L(R))
  • The Core Model Program and Determinacy

Part III: Forcing and Forcing Axioms

• Chapter 6: Introduction to Forcing
  • Generic Extensions and Boolean-Valued Models
  • Partial Orders and Generic Filters
  • Forcing Notions and Generic Objects (e.g., Cohen Forcing)
  • Forcing over L and Independence Results
• Chapter 7: Martin's Axiom (MA) and Its Consequences
  • Definition of Martin's Axiom
  • Properties of Partial Orders (Countable Chain Condition, Properness)
  • Consequences of MA for Cardinal Arithmetic (e.g., 2ℵ0​=ℵ1​)
  • Applications in Topology and Measure Theory
• Chapter 8: Strong Forcing Axioms
  • Proper Forcing Axiom (PFA): Definition and Context
  • Semiproper Forcing Axiom (SPFA) and Bounded Forcing Axiom (BFA)
  • Consequences of PFA for 2ℵ0​ and Higher Cardinals
  • Applications in Topology (e.g., Normal Moore Space Conjecture) and Set Theory

Part IV: The Nonstationary Ideal Under Axiomatic Assumptions

• Chapter 9: The Nonstationary Ideal and Determinacy
  • Properties of the Nonstationary Ideal under AD
  • Measures and Filters on ω1​ under AD
  • The Ideal of Sets of Ordinals Not Containing a Club
• Chapter 10: The Nonstationary Ideal and Forcing Axioms
  • Properties of the Nonstationary Ideal under MA and PFA
  • Reflection Principles and Forcing Axioms
  • The Failure of ♢ and □ under Forcing Axioms
  • Distinguishing Forcing Axioms by their Impact on Ideals

Part V: Incompatibilities, Consistency, and Advanced Topics

• Chapter 11: Incompatibilities and Interactions
  • The Fundamental Conflict Between AD and Standard Forcing Axioms
  • Consistency of Combinations (e.g., AD and its weaker forms, or FAs and large cardinals)
  • The Role of Inner Models in Reconciling Axioms
• Chapter 12: Generalizations and Open Problems
  • Determinacy for Larger Classes of Sets (e.g., Projective Determinacy)
  • The Axiom of Choice and Determinacy
  • Recent Developments in Determinacy and Forcing Axioms
  • Key Open Problems and Research Directions

People also search for The axiom of determinacy forcing axioms and the nonstationary ideal 1st:

the axiom of determinacy forcing axioms and the nonstationary ideal
    
can axioms be proven
    
what are axioms examples
    
how do we know axioms are true
    
what are the 5 axioms

Tags: W Hugh Woodin, axiom, determinacy