Ergodicity as an obstruction to classification by higher-rank invariants

Given a collection of mathematical objects and a natural notion of equivalence (such as isomorphism), we want to find a complete classification. This paper presents methods for proving anti-classification results, that is, when a complete classification is not possible using certain classifying invariants. The problem of using real numbers as classifying invariants is well understood. In this case, the obstruction to classifiability is ergodicity. We consider classifiability by higher rank objects, such as countable sets of real numbers, and present a simple ergodicity-like obstruction for classifiability. This method is particularly useful when applied in various models of ZF where even weak fragments of the axiom of choice fail.