Finally, we define the theories \(\mathsf{S}^i_2 = \mathsf{BASIC} + \Sigma^b_i\)-\(\text{PIND}\) and \(\mathsf{T}^i_2 = \mathsf{BASIC} + \Sigma^b_i\)-\(\text{IND}\) and also \(\mathsf{S}_2 = \bigcup_{i \in \mathbb{N}} \mathsf{S}^i_2\) and \(\mathsf{T}_2 = \bigcup_{i \in \mathbb{N}} \mathsf{T}^i_2\).