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Generalized Descriptive Set Theory and Classification Theory
Sy-David Friedman, Kurt Gödel Research Center, Vienna, Austria, Tapani Hyttinen, University of Helsinki, Finland, and Vadim Kulikov, Kurt Gödel Research Center, Vienna, Austria
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Memoirs of the American Mathematical Society
2014; 80 pp; softcover
Volume: 230
ISBN-10: 0-8218-9475-7
ISBN-13: 978-0-8218-9475-0
List Price: US$65
Individual Members: US$39
Institutional Members: US$52
Order Code: MEMO/230/1081
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Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper the authors study the generalization where countable is replaced by uncountable. They explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. They also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. The authors' results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.

Table of Contents

  • History and motivation
  • Introduction
  • Borel sets, \(\Delta_1^1\) sets and infinitary logic
  • Generalizations from classical descriptive set theory
  • Complexity of isomorphism relations
  • Reductions
  • Open questions
  • Bibliography
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