 We shall not give a detailed account of the formalization of these theories, but focus on the non-iterated case \(\bID_1\) and its intuitionistic version \(\bID_1^i\) to convey the idea. A monotone operator on \(\bbN\) is a map \(\Gamma\) that sends a set \(X\subseteq \bbN\) to a subset \(\Gamma(X)\) of \(\bbN\) and is monotone, i.e., \(X\subseteq Y\subseteq \bbN\) implies \(\Gamma(X) \subseteq \Gamma(Y)\). Owing to monotonicity, the operator \(\Gamma\) will have a least fixed point \(I_{\Gamma}\subseteq\bbN\), i.e., \(\Gamma( I_{\Gamma})=I_{\Gamma}\) and for every other fixed point X it holds \(I_{\Gamma} \subseteq X\). Set-theoretically \(I_{\Gamma}\) is obtained by iterating \(\Gamma\) through the ordinals, 