# Can extensions of $Q$ contradict Löb with recursive reflection?

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?

• Is it true that for every sentence $\phi$, $Q\vdash Pr_Q(\ulcorner \bot\urcorner)\to Pr_Q(\ulcorner \phi\urcorner)$ ? If that's the case then Lob's theorem is provable in $Q$. May 1 '17 at 9:04
• What is a "recursive sentence"? May 1 '17 at 14:00
• @NoahSchweber Here I take a recursive sentence to be a $\Delta_1$-sentence. Is that not common?
– FAB
May 1 '17 at 14:56
• I haven't personally seen it used that way, but that doesn't mean it's not common; I just didn't know what you meant. May 1 '17 at 16:17
• I see now that there is different usage. Here is one article which talks about $\Delta_1$ formulas: en.wikipedia.org/wiki/Reverse_mathematics#The_base_system_RCA0. But now that you bring it up I see that it is probably best to use $\Delta_{n}$ only for sets defined both by $\Sigma{n}$ and $\Pi_{n}$ formulas, as the criteria for identifying formulas are purely syntactical.
– FAB
May 1 '17 at 16:31

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.)