 In the light of the above theorem second-order logic sits firmly on the \(\Sigma_2\cup\Pi_2\)-level of set theoretic definability. This is a reason to consider second-order logic weaker than first order set theory. This sounds paradoxical. How can second-order logic, with all the power and non-first order properties it has, be weaker than first order logic based set theory, when first order logic cannot even express finiteness, countability, uncountability, and so on? Nevertheless the message of  Theorem 4  is undeniable. Perhaps this would not be so puzzling if we thought of set theory as a very high order logic over the singleton of the empty set. After all, set theory permits endless iterations of the power set operation, while second-order logic permits only one iteration.