I'm trying to understand the inapproximability of the minimum set cover problem.
The wikipedia page states that it is hard to approximate within a factor of $(1-o(1))\ln n$ and that $n$ refers to the size of the ground set. It references the work of Dinur and Steurer (2013) who state that, for every $\varepsilon>0$, it is NP-hard to approximate min set cover within a factor of $(1-\varepsilon)\ln n$ where $n$ refers to the size of the instance. It seems they don't define what they mean by "size of the instance," but I take that to mean an encoding of the instance (which takes more space than just the size of the ground set).
Would I be correct in saying that set cover is hard to approximate within a factor of $(1-\varepsilon)\ln N$, where $N$ is the size of the ground set plus the sizes of all the subsets? Or, if $N$ is the size of the ground set plus the number of subsets?