# is the Hyperarithmetical hierarchy second order arithmetic but NOT second order logic? and if so why?

I was told in a previous answer (but dont remember by whom but he then didnt answer back), that the hyperarithmetical hierarchy is second order arithmetic but not second order logic. Is this so? what precludes to interpret those formulas with full semantics? do the sets defined by a given statement differ with the two different semantics (full vs Henkin) ) or they are the same? The motivation for this question has to do with some more complicated question that I plan to make in the short future

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Do you mean the analytical hierarchy (e.g $\Sigma^1_4$, $\Pi^1_8$)? The hyperarithmetical hierarchy is about transfinitely iterated Turing jumps, and as such it only classifies sets that are already $\Delta^1_1$. –  Carl Mummert Feb 27 '13 at 12:37
My concern is related to the analytical hierarchy, but the specific response was on the hyperaritmethical ones, that is why I limited the question to it. –  Wolphram jonny Feb 27 '13 at 13:25