Uncountable free abelian groups via κ-computability
Linda Brown Westrick
University of Connecticut
One way to study structures of uncountable cardinality κ is to generalize the notion of computation. Saying that a subset of κ is κ-c.e. if it is Σ01 definable (with parameters, in the language of set theory) over Lκ provides the notion of κ-computability. We may also quantify over subsets of Lκ, providing a notion of a κ-analytic set (here we assume V=L). In this setting, we consider the difficulty of recognizing free groups and the complexity of their bases. For example, if κ is a successor cardinal, the set of free abelian groups of size κ is Σ11-complete. If κ is the successor of a regular cardinal which is not weakly compact, there is a computable free abelian group of cardinality κ, all of whose bases compute ∅”, and this is the best coding result possible. The resolution of questions of this type is more complex for other κ, and a few questions remain open. This is joint work with Greenberg and Turetsky.
Linda Brown Westrick received her doctorate in 2014 from the University of California at Berkeley, under the supervision of Ted Slaman. Currently she holds a postdoctoral position at the University of Connecticut. She works in computability theory and effective descriptive set theory, applying techniques from these areas to questions in analysis, symbolic dynamics, and chaos.