If I understood the article correctly (and I think I didn't) it says learnability of a given data-set is like the halting problem, there is no general algorithm for deciding whether an arbitrary data-set is learnable. Am I close?
This was my takeaway from the article as well. Even though I have no academic knowledge on the subject, can't say I was surprised to learn this. To me it seems obvious in hindsight, though might be Dunning-Kruger.