Decidable language that has no finite description?
Unless you want to be more rigid in what you allow as a &quot;description&quot;, every decidable language has a finite description - precisely the Turing machine that decides it. This Turing machine must exist (otherwise the language would not be decidable).
Although we often thing of automata as machines that do work, mathematically they're just a way of, in a certain sense, compressing the table that says &quot;yes&quot; or &quot;no&quot; for every string (over the given alphabet).