TITLE: proof theoretic ordinal for Robinson's arithmetic
QUESTION [7 upvotes]: Does a theory like Robinson's arithmetic have a proof-theoretic ordinal? If so, what is it?

REPLY [3 votes]: 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.
(This is in response to a side question that came up in the comment thread of Noah's answer, but I don't have the karma to post this as a comment).