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.
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