The present work may perhaps be seen as a point of convergence of two historically distinct sequences of results. One sequence of results started with the work of Tennenbaum [59] who showed that there could be no nonstandard recursive model of the system PA of first order Peano arithmetic. Shepherdson [65] on the other hand showed that the system of arithmetic with open induction was sufficiently weak to allow the construction of nonstandard recursive models. Between these two results there remained for many years a large gap occasioned by a general lack of interest in weak systems of arithmetic. However Dana Scott observed that the addition alone of a nonstandard model of PA could not be recursive, while more recently McAloon [82] improved these results by showing that even for the weaker system of arithmetic with only bounded induction, neither the addition nor the multiplication of a nonstandard model could be recursive.Another sequence of results starts with the work of Lessan [78], and independently Jensen and Ehrenfeucht [76], who showed that the structures which may be obtained as the reducts to addition of countable nonstandard models of PA are exactly the countable recursively saturated models of Presburger arithmetic. More recently, Cegielski, McAloon and the author [81] showed that the above result holds true if PA is replaced by the much weaker system of bounded induction.However in both the case of the Tennenbaum phenomenon and in that of the recursive saturation of addition the problem remained open as to how strong a system was really necessary to generate the required phenomenon. All that was clear a priori was that open induction was too weak to produce either result.

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The present paper seeks to establish a logical foundation for studying axiomatically multi-agent probabilistic reasoning over a discrete space of outcomes. We study the notion of a social inference process which generalises the concept of an inference process for a single agent which was used by Paris and Vencovská to characterise axiomatically the method of maximum entropy inference. Axioms for a social inference process are introduced and discussed, and a particular social inference process called the Social Entropy Process, or SEP, is defined which satisfies these axioms. SEP is justified heuristically by an information theoretic argument, and incorporates both the maximum entropy inference process for a single agent and the multi-agent normalised geometric mean pooling operator.

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