 It is again possible to prove  Theorem 3.10  by a direct syntactic construction. For instance, building on the definition of the universal \(\Sigma^0_1\)-predicate \(\sigma_{k,1}(\vec{y})\), it may be shown that for every level \(\Sigma^0_n\) of the arithmetical hierarchy, there is a \(\Sigma^0_n\)-formula \(\sigma_{k,n}(x,\vec{y})\) which defines \(\Sigma^0_n\)-satisfaction in the standard model in the sense that 