**Memoirs of the American Mathematical Society**

2004;
141 pp;
Softcover

MSC: Primary 03;

Print ISBN: 978-0-8218-3450-3

Product Code: MEMO/167/793

List Price: $68.00

AMS Member Price: $40.80

MAA Member Price: $61.20

**Electronic ISBN: 978-1-4704-0391-1
Product Code: MEMO/167/793.E**

List Price: $68.00

AMS Member Price: $40.80

MAA Member Price: $61.20

# Descriptive Set Theory and Definable Forcing

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*Jindřich Zapletal*

The subject of the book is the relationship between definable forcing and descriptive set theory. The forcing serves as a tool for proving independence of inequalities between cardinal invariants of the continuum. The analysis of the forcing from the descriptive point of view makes it possible to prove absoluteness theorems of the type “certain forcings are the provably best attempts to achieve consistency results of certain syntactical form” and others. There are connections to such fields as pcf theory, effective descriptive set theory, determinacy and large cardinals, Borel equivalence relations, abstract analysis, and others.

#### Readership

Graduate students and research mathematicians interested in mathematical logic and foundations.

#### Table of Contents

# Table of Contents

## Descriptive Set Theory and Definable Forcing

- Contents v6 free
- 1 Introduction 110 free
- 2 Definable forcing adding a single real 514
- 2.1 The factor algebras 514
- 2.2 Basic descriptive set theoretic considerations 918
- 2.3 Examples 1423
- 2.3.1 The ideal of countable sets 1524
- 2.3.2 The ideal of σ…bounded sets 1524
- 2.3.3 The ideal of meager sets 1625
- 2.3.4 The cmin ideal 1625
- 2.3.5 Ideals generated by closed sets 1928
- 2.3.6 The Laver ideal 2130
- 2.3.7 Ideals associated with creature forcings 2231
- 2.3.8 The Lebesgue null ideal 2433
- 2.3.9 Mathias forcing 2433
- 2.3.10 The E[sub(0)] ideal 2534
- 2.3.11 Silver forcing 2938
- 2.3.12 The σ…porous ideal 3140
- 2.3.13 Steprans forcing 3342
- 2.3.14 Hausdorff measures 3746
- 2.3.15 Unions of ideals 4150
- 2.3.16 Cross–products of ideals 4251
- 2.3.17 The σ…splitting ideal 4554
- 2.3.18 Namba forcing 4554

- 3 The countable support iteration 4756
- 4 Other forcings 7180
- 5 Applications 91100
- A: Examples of cardinal invariants 117126
- B: The syntax of cardinal invariants 119128
- C: Effective descriptive set theory 125134
- D: Large cardinals 133142