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](https://mathoverflow.net/questions/233659/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?



Can extensions of $Q$ contradict Löb with recursive reflection?
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.)


