 There is a sense in which \(\Pi^1_1\), or equivalently \(\Sigma^1_1\), already has the full power of second-order logic, although the above hierarchy result (\ref{kurat}) shows that this cannot literally be true. To see this, i.e., to see what the reduction of second-order logic to its \(\Pi^1_1\)-part means, let L be a vocabulary, E a binary and U a unary relation symbol, both not in L. Let \(\theta\) be the second-order \(\Pi^1_1\)-formula 