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
set-theoretic 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.
Readership
Graduate students and research mathematicians interested in set
theory. especially descriptive set theory.