 To summarize: when it is said, in the context of the incompleteness theorems, that “a certain amount of elementary arithmetic can be carried out” in a system, this usually means that it contains PRA or at least Q. For the first incompleteness theorem, Q is sufficient; for the standard proofs of the second theorem, something like PRA, at a minimum, is needed. There is a version of the second incompleteness theorem for Q (see Bezboruah &amp; Shepherdson 1976) but there has been some debate on whether the relevant statement in Q can really be taken to express consistency, Q being so weak (see Kreisel 1958; Bezboruah &amp; Shepherdson 1976; Pudlák 1996; Franks 2009).