Notes on Forcing Axioms

In the mathematical practice, the Baire category method is a tool for establishing the existence of a rich array of generic structures. However, in mathematics, the Baire category method is also behind a number of fundamental results such as the Open Mapping Theorem or the Banach–Steinhaus Boundedness Principle. This volume brings the Baire category method to another level of sophistication via the internal version of the set-theoretic forcing technique. It is the first systematic account of applications of the higher forcing axioms with the stress on the technique of building forcing notions rather than on the relationship between different forcing axioms or their consistency strengths.

Contents:

• Baire Category Theorem and the Baire Category Numbers

• Coding Sets by the Real Numbers

• Consequences in Descriptive Set Theory

• Consequences in Measure Theory

• Variations on the Souslin Hypothesis

• The S-Spaces and the L-Spaces

• The Side-condition Method

• Ideal Dichotomies

• Coherent and Lipschitz Trees

• Applications to the S-Space Problem and the von Neumann Problem

• Biorthogonal Systems

• Structure of Compact Spaces

• Ramsey Theory on Ordinals

• Five Cofinal Types

• Five Linear Orderings

• Cardinal Arithmetic and mm

• Reflection Principles

• Appendices:

  • Basic Notions
  • Preserving Stationary Sets
  • Historical and Other Comments

Readership: Graduate students and researchers in logic, set theory and related fields.
Key Features:

• This is a first systematic exposition of the unified approach for building proper, semi-proper, and stationary preserving forcing notions through the method of using elementary submodels as side conditions
• The books starts from the classical applications of Martin's axioms and ends with some of the most sophisticated applications of the Proper Forcing Axioms. In this way, the reader is led into a natural process of understanding the combinatorics hidden behind the method