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This book introduces a new research direction in set theory: the
study of models of set theory with respect to their extensional
overlap or disagreement. In Part I, the method is applied to
isolate new distinctions between Borel equivalence relations. Part
II contains applications to independence results in
Zermelo-Fraenkel set theory without Axiom of Choice. The method
makes it possible to classify in great detail various paradoxical
objects obtained using the Axiom of Choice; the classifying
criterion is a ZF-provable implication between the existence of
such objects. The book considers a broad spectrum of objects from
analysis, algebra, and combinatorics: ultrafilters, Hamel bases,
transcendence bases, colorings of Borel graphs, discontinuous
homomorphisms between Polish groups, and many more. The topic is
nearly inexhaustible in its variety, and many directions invite
further investigation.
This book provides an introduction to axiomatic set theory and
descriptive set theory. It is written for the upper level
undergraduate or beginning graduate students to help them prepare
for advanced study in set theory and mathematical logic as well as
other areas of mathematics, such as analysis, topology, and
algebra.The book is designed as a flexible and accessible text for
a one-semester introductory course in set theory, where the
existing alternatives may be more demanding or specialized. Readers
will learn the universally accepted basis of the field, with
several popular topics added as an option. Pointers to more
advanced study are scattered throughout the text.
This book provides an introduction to mathematical logic and the
foundations of mathematics. It will help prepare students for
advanced study in set theory and mathematical logic as well as
other areas of mathematics, such as analysis, topology, and
algebra. The presentation of finite state and Turing machines leads
to the Halting Problem and Goedel's Incompleteness Theorem, which
have broad academic interest, particularly in computer science and
philosophy.
This volume presents some exciting new developments occurring on
the interface between set theory and computability as well as their
applications in algebra, analysis and topology. These include
effective versions of Borel equivalence, Borel reducibility and
Borel determinacy. It also covers algorithmic randomness and
dimension, Ramsey sets and Ramsey spaces. Many of these topics are
being discussed in the NSF-supported annual Southeastern Logic
Symposium.
This book lays the foundations for an exciting new area of research
in descriptive set theory. It develops a robust connection between
two active topics: forcing and analytic equivalence relations. This
in turn allows the authors to develop a generalization of classical
Ramsey theory. Given an analytic equivalence relation on a Polish
space, can one find a large subset of the space on which it has a
simple form? The book provides many positive and negative general
answers to this question. The proofs feature proper forcing and
Gandy-Harrington forcing, as well as partition arguments. The
results include strong canonization theorems for many classes of
equivalence relations and sigma-ideals, as well as ergodicity
results in cases where canonization theorems are impossible to
achieve. Ideal for graduate students and researchers in set theory,
the book provides a useful springboard for further research.
Descriptive set theory and definable proper forcing are two areas
of set theory that developed quite independently of each other.
This monograph unites them and explores the connections between
them. Forcing is presented in terms of quotient algebras of various
natural sigma-ideals on Polish spaces, and forcing properties in
terms of Fubini-style properties or in terms of determined infinite
games on Boolean algebras. Many examples of forcing notions appear,
some newly isolated from measure theory, dynamical systems, and
other fields. The descriptive set theoretic analysis of operations
on forcings opens the door to applications of the theory:
absoluteness theorems for certain classical forcing extensions,
duality theorems, and preservation theorems for the countable
support iteration. Containing original research, this text
highlights the connections that forcing makes with other areas of
mathematics, and is essential reading for academic researchers and
graduate students in set theory, abstract analysis and measure
theory.
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