# Derivability conditions for Robinson arithmetic

Two pieces of hearsay I have encountered about Robinson's Q:

1. Q fails to satisfy the Löb derivability conditions;
2. Pudlák criticised the Löb derivability conditions and suggested rival, weaker conditions.

Which leads to three questions, if the above are right:

1. Which derivability condition(s) does Q not satisfy;
2. What were Pudlák's rival conditions and what was his complaint with the Löb conditions; and
3. Does Q satisfy the rival conditions?

These questions arose from Carl Mummert's answer to a math.sx question of mine, Can Robinson's Q prove Presburger arithmetic consistent?.

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I can take blame for passing along part (1) of the hearsay. When I went back to look for where I had seen it, I found published claims that Q is "too weak" to prove the derivability conditions, but not an explicit example nor a reference. Perhaps those authors were only referring to the fact that the usual proofs of the derivability conditions don't go through in Q, and I read too much into their statements. So I'm also interested to learn whether there is a published counterexample to the provability of the derivability conditions in Q. –  Carl Mummert Sep 15 '10 at 21:42

First, note than one can interpret Sam Buss's bounded arithmetic theories like $S^1_2$ in $Q$, so it is not as weak as it seems at first sight in expressing and proving theorems. One can use a reasonable formula to exclude those non-standard numbers which are too pathological and prove consistency of $L$ (if $S^1_2$ proves consistency of $L$).
(By the way, Bezboruah et al. (1976) seems to be a decade before Nelson's Predicative Arithmetic (1986) which shows that one can interpret $I\Delta_0$ in Q.)