Consider the hierarchy given by $\cal S\_0 =$ first-order Peano arithmetic, $\cal S\_{\alpha+1}=\cal S\_{\alpha} + Con(S\_\alpha)$ (a consistency statement for $\cal S\_\alpha$), and if $\alpha$ is a limit ordinal, $\cal S\_\alpha=\bigcup\_{\beta < \alpha} \cal S\_\beta$. 


Peano arithmetic proves transfinite induction for ordinals below $\epsilon\_0$. Is there an $\alpha$ such that $\cal S\_\alpha$ proves transfinite induction for longer initial segments? If so, is there a fixed point $\alpha$ s.t. $\epsilon\_0 < \alpha \leq \Gamma\_0$ and $\cal S\_\alpha$ proves transfinite induction for all ordinals below $\alpha$ but not beyond?



Adding consistency statements to Peano arithmetic allows more instances of transfinite induction?
Over PRA, the transfinite induction schema up to $\epsilon\_0$ for primitive recursive formulas implies not just the consistency of PA, but also its $\Sigma^0\_1$-soundness (i.e., the uniform $\Sigma^0\_1$-reflection principle for PA). As such, it is not provable in any consistent extension of PA by a set of $\Pi^0\_1$ sentences, because of Gödel’s theorem.


