The theories \(\mathsf{S}^i_2\) and \(\mathsf{T}^i_2\) both extend a base theory known as \(\textsf{BASIC}\). \(\textsf{BASIC}\) contains 32 axioms, which include those of \(\mathsf{Q}\), as well as others which axiomatize the intended interpretations of \(\lvert \dots\rvert\), \(\lfloor \frac{x}{2} \rfloor\) and \(\#\) (see, e.g., Hájek and Pudlák 1998, 321–22). We also define the following induction schemas: