Hardcover ISBN:  9780821848135 
Product Code:  SURV/155 
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Electronic ISBN:  9781470413828 
Product Code:  SURV/155.E 
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Book DetailsMathematical Surveys and MonographsVolume: 155; 2009; 502 ppMSC: Primary 03; Secondary 28; 54;
Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets. This subject was started by the French analysts at the turn of the 20th century, most prominently Lebesgue, and, initially, was concerned primarily with establishing regularity properties of Borel and Lebesgue measurable functions, and analytic, coanalytic, and projective sets. Its rapid development came to a halt in the late 1930s, primarily because it bumped against problems which were independent of classical axiomatic set theory. The field became very active again in the 1960s, with the introduction of strong settheoretic hypotheses and methods from logic (especially recursion theory), which revolutionized it.
This monograph develops Descriptive Set Theory systematically, from its classical roots to the modern “effective” theory and the consequences of strong (especially determinacy) hypotheses. The book emphasizes the foundations of the subject, and it sets the stage for the dramatic results (established since the 1980s) relating large cardinals and determinacy or allowing applications of Descriptive Set Theory to classical mathematics.
The book includes all the necessary background from (advanced) set theory, logic and recursion theory.ReadershipGraduate students and research mathematicians interested in set theory. especially descriptive set theory.

Table of Contents

Chapters

1. Introduction

2. The basic classical notions

3. $\kappa $Suslin and $\lambda $Borel

4. Basic notions of the effective theory

5. Structure theory for pointclasses

6. The constructible universe

7. The playful universe

8. The recursion theorem

9. Metamathematics


Additional Material

Reviews

The author added the most important references to the developments in descriptive set theory since 1980 when they touch questions formulated in the book.
Zentralblatt MATH


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Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets. This subject was started by the French analysts at the turn of the 20th century, most prominently Lebesgue, and, initially, was concerned primarily with establishing regularity properties of Borel and Lebesgue measurable functions, and analytic, coanalytic, and projective sets. Its rapid development came to a halt in the late 1930s, primarily because it bumped against problems which were independent of classical axiomatic set theory. The field became very active again in the 1960s, with the introduction of strong settheoretic hypotheses and methods from logic (especially recursion theory), which revolutionized it.
This monograph develops Descriptive Set Theory systematically, from its classical roots to the modern “effective” theory and the consequences of strong (especially determinacy) hypotheses. The book emphasizes the foundations of the subject, and it sets the stage for the dramatic results (established since the 1980s) relating large cardinals and determinacy or allowing applications of Descriptive Set Theory to classical mathematics.
The book includes all the necessary background from (advanced) set theory, logic and recursion theory.
Graduate students and research mathematicians interested in set theory. especially descriptive set theory.

Chapters

1. Introduction

2. The basic classical notions

3. $\kappa $Suslin and $\lambda $Borel

4. Basic notions of the effective theory

5. Structure theory for pointclasses

6. The constructible universe

7. The playful universe

8. The recursion theorem

9. Metamathematics

The author added the most important references to the developments in descriptive set theory since 1980 when they touch questions formulated in the book.
Zentralblatt MATH