On Q

In this paper we study the theory Q. We prove a basic result that says that, in a sense explained in the paper, Q can be split into two parts. We prove some consequences of this result. (i) Q is not a poly-pair theory. This means that, in a strong sense, pairing cannot be defined in Q. (ii) Q does not have the Pudlák Property. This means that there two interpretations of S12 in Q which do not have a definably isomorphic cut. (iii) Q is not sententially equivalent with PA−. This tells us that we cannot do much better than mutual faithful interpretability as a measure of sameness of Q and PA−. We briefly consider the idea of characterizing Q as the minimal-in-some-sense theory of some kind modulo some equivalence relation. We show that at least one possible road towards this aim is closed.