Generalized descriptive set theory and classification theory

Source abstract

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We 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. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations. Acknowledgement : The authors wish to thank the John Templeton Foundation for its generous support through its project Myriad Aspects of Infinity (ID #13152). The authors wish to thank also Mittag-Leffler Institute (the Royal Swedish Academy of Sciences). The second and the third authors wish to thank the Academy of Finland for its support through its grant number 1123110. The third author wants to express his gratitude to the Research Foundation of the University of Helsinki and the Finnish National Graduate School in Mathematics and its Applications for the financial support during the work. We are grateful to Jouko Väänänen for the useful discussions and comments he provided on a draft of this paper.