10,756 research outputs found
Generalized descriptive set theory and classification theory
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
Invariant measures concentrated on countable structures
Let L be a countable language. We say that a countable infinite L-structure M
admits an invariant measure when there is a probability measure on the space of
L-structures with the same underlying set as M that is invariant under
permutations of that set, and that assigns measure one to the isomorphism class
of M. We show that M admits an invariant measure if and only if it has trivial
definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary
finite tuple of M fixes no additional points. When M is a Fraisse limit in a
relational language, this amounts to requiring that the age of M have strong
amalgamation. Our results give rise to new instances of structures that admit
invariant measures and structures that do not.Comment: 46 pages, 2 figures. Small changes following referee suggestion
Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals
We describe the basic theory of infinite time Turing machines and some recent
developments, including the infinite time degree theory, infinite time
complexity theory, and infinite time computable model theory. We focus
particularly on the application of infinite time Turing machines to the
analysis of the hierarchy of equivalence relations on the reals, in analogy
with the theory arising from Borel reducibility. We define a notion of infinite
time reducibility, which lifts much of the Borel theory into the class
in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference,
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