 Given a natural ordinal representation system \(\langle A,\prec,\ldots\rangle\) of order type \(\tau\) let \(\PRA+\rTI_{\qf}(&lt;\tau)\) be PRA augmented by quantifier-free induction over all initial (externally indexed) segments of \(\prec\). This is perhaps best explained via the representation system for \(\varepsilon_0\) given above. There one can take the initial segments of \(\prec\) to be determined by the Gödel numbers of the ordinals \(\omega_0\coloneqq 1\) and \(\omega_{n+1}\coloneqq \omega^{\omega_n}\) whose limit is \(\varepsilon_0\).