Structures interpretable in models of bounded arithmetic

We look for a converse to a result from [N. Thapen, A model-theoretic characterization of the weak pigeonhole principle, Annals of Pure and Applied Logic 118 (2002) 175–195] that if the weak pigeonhole principle fails in a model K of bounded arithmetic, then there is an end-extension of K interpretable inside K . We show that if a model J of an induction-free theory of arithmetic is interpretable inside K , then either J is isomorphic to an initial segment of K ( J is “smaller” than K ), or K is isomorphic to an initial segment of J ( J is “bigger” than K ) and in this case the weak pigeonhole principle fails in K . This result is formulated in terms of a theory S 0 1 of bounded arithmetic with a greatest element. on to consider structures defined by oracles, and use the probabilistic witnessing theorem for S 2 1 + ( dual WPHP ( PV ) ) to give a general criterion for what can be proved about these using the weak pigeonhole principle. We also show that the injective WPHP is not provable in this theory in the relativized case.