As I have not specified I, you need to show that my choice of S_I is inconsistent with any possible set I in order to make the argument you’re trying to make.
In the original paper, S_I is determined by I. There is no I for which your set S_I is the set defined in the paper. It really is as simple as that.
In the original paper, S_I is determined by I. There is no I for which your set S_I is the set defined in the paper. It really is as simple as that.