I'm quite sure I don't understand very well the links between proof theoretical ordinals of theories, the axioms of transfinite induction and the objects a theory can prove to exist.

For instance I'm considering *Peano Axioms* ($\mathbf{PA}$), of proof theoretic ordinal $\epsilon_0$, *Primitive Recursive Arithmetic* ($\mathbf{PRA}$) of proof theoretic ordinal $\omega^\omega$ and *Elementary Recursive Arithmetic* ($\mathbf{ERA}$), which is a fragment of $\mathbf{PRA}$.

I was wondering if $\mathbf{PRA}+TI\{\alpha\in\epsilon_0\}$ (where $TI$ stands for transfinite induction) was equivalent in some sense to $\mathbf{PA}+TI\{\alpha\in\epsilon_0\}$ or/and to $\mathbf{ERA}+TI\{\alpha\in\epsilon_0\}$ ?

And more generally, if it was true that for any set $A$ of (countable) ordinals such that $\epsilon_0 \subset A$, $\mathbf{ERA}+TI\{\alpha\in A\} = \mathbf{PRA}+TI\{\alpha\in A\} = \mathbf{PA}+TI\{\alpha\in A\}$ ?

Any enlightenment would be most welcome :) Thanks in advance.