The authors desire to characterize the natural numbers by their structure, as opposed to designating a particular choice of objects like \(\emptyset\), \(\{\emptyset\}\), etc. But at the same time, they wish to rule out nonstandard models -- and not to make their description circular. To accomplish this, they invoke Tennenbaum's Theorem and propose, ``Intended models are notation systems with recursive operations on them satisfying the Peano axioms.'' Here, to avoid circularity, they take ``recursive'' in an informal sense, appealing to Church's Thesis.