# proof theoretic ordinal for Robinson's arithmetic

Does a theory like Robinson's arithmetic have a proof-theoretic ordinal? If so, what is it?

Well, since $Q$ doesn't even have induction along $\mathbb{N}$, I'm dubious that the notion of proof-theoretic ordinal makes sense for it; but if forced, I'd say the answer has to be $\omega$. Induction along finite orderings is trivial, so $\omega$ is the first ordinal for which it makes sense to ask "Does $Q$ prove induction along this ordinal?," and $Q$ doesn't.
• Well, I guess I was hoping for some ordinal $\alpha$ such that the principlw of transfinite induction up to $\alpha$ plus some minimal theory (e.g., PRA) proves Con(Q). Oct 20 '16 at 3:10
• On the other hand, the proof theoretic ordinal of PRA is $\omega^\omega$, while the proof theoretic ordinal of EFA is just $\omega^3$. So that suggests that PRA proves Con(EFA) and hence Con(Q) . . . Oct 20 '16 at 3:46
Noah is correct that PRA $$\vdash$$ Con(Q): On p. 139 of the Handbook of Proof Theory, Chapter 2 (available here), Buss shows that $$I\Delta_0$$ + "tetration is total" $$\vdash$$ Con($$I\Delta_0$$), so it certainly proves Con(Q). Since $$I\Delta_0 \subseteq$$ PRA and PRA proves tetration total, the result follows.