The generic Muchnik degrees, introduced by Schweber, give a way of comparing the computability-theoretic content of uncountable structures. Though obscured slightly by the need for some set-theoretic machinery, I hope to highlight how this notion really gives an easy and natural way to talk about computable structure theory for uncountable structures. I will focus on the tool of complexity profiles.
Complexity profiles are a way of measuring, for two structures A generic Muchnik reducible to B, which subsets of A can be defined using B. The complexity profile of A on itself is the natural analog of considering the relatively intrinsically Sigma_k sets in A.
Using complexity profiles, I will compare three generic Muchnik degrees: Cantor space < Baire space < the Borel-complete degree. In particular, I will describe some dichotomy theorems regarding simple expansions of these and describe how to build degrees strictly between them. (Joint work with Joseph S. Miller, Noah Schweber, and Mariya Soskova.)