TITLE: Can extensions of $Q$ contradict Löb with recursive reflection?
QUESTION [5 upvotes]: It is an odd and arguably unacceptable situation that $PA$ does not have   $\vdash_{PA}(Pr_{PA}\ulcorner A\urcorner\to A)$  for false recursive sentences $A$. 
However, it is not clear to me that Löb's theorem is already derivable in Robinson arithmetic $Q$, for one cannot assume that the provability predicate of $Q$ obeys all the Löb derivability conditions. (Compare to these matter question A question on the provability predicate of Q).
Are there natural omega consistent extensions $Q*$ of $Q$ such  $\vdash_{Q*}(Pr_{Q*}\ulcorner A\urcorner\to A)$ for all $\Delta_1$ sentences?

REPLY [6 votes]: No consistent recursively axiomatized extension $T$ of $Q$ can prove $\mathrm{Pr}_T\ulcorner\bot\urcorner\to\bot$, that is, $\mathrm{Con}_T$.
In fact, no consistent r.e. theory $T$ can interpret $Q+\mathrm{Con}_T$.
In fact, no consistent r.e. theory $T$ can interpret $Q+\{\mathrm{RCon}_T(\overline n):n\in\mathbb N\}$, where $\mathrm{RCon}_T(x)$ denotes the consistency of $T$ with respect to proofs using only formulas of “complexity” $x$. See Pudlák’s Cuts, Consistency Statements and Interpretations. (He proves it for interpretations on a cut, but this does not really make a difference.)