Indiscernibility of indiscernibles in second order logic

Question

It is not clear to me if the statements in $0^\#$ remain indiscernible when we move to second order logic. Or are there second logic formulas that can discriminate between first order indiscernibles?

Answer 1

I assume that you intend second-order formulas to be interpreted with the first-order variables ranging over the constructible sets (since that's where you have first-order indiscernibility) and the second-order variables ranging over arbitrary (not necessarily constructible) subsets (or maybe subclasses) of L, i.e., the standard interpretation of second-order logic over L. If that's your intention, then the Silver indiscernibles are not second-order indiscernible. For example, second-order logic over L can express "$x$ is a cardinal (in the sense of the full universe, not just in L)" and this property is true of some but not all of the Silver indiscernibles.