Once we accept as logically sound the set-theoretically based meta-argumentation that the first-order Peano Arithmetic PA can be forced to admit set-theoretical models which contain elements other than the natural numbers (where the meta-theory is taken to be a set-theory such as ZF or ZFC, and the logical consistency of the meta-theory is not considered relevant to the argumentation), then the properties of the algebraic and arithmetical structures of such putative models (as detailed, for instance, in standard texts such as Richard Kayes ‘Models of Peano Arithmetic’) should perhaps follow without serious foundational reservation.
However in a recent preprint I argue that, if we scrupulously avoid implicit appeal to any non-constructive considerations then, even from a classical perspective, there are serious foundational reservations to accepting that a consistent PA can be forced to admit non-standard models which contain elements other than the natural numbers.
 Andrey I. Bovykin. 2000. On order-types of models of arithmetic. Thesis submitted to the University of Birmingham for the degree of Ph.D. in the Faculty of Science, School of Mathematics and Statistics, The University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom, 13 April 2000.
 George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. Computability and Logic (4th ed). Cambridge University Press, Cambridge, ch.25, p.302.
 Richard Kaye. 1991. Models of Peano Arithmetic. Oxford Logic Guides, 15. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1991, p.1.
 Richard Kaye. 2011. Tennenbaum’s theorem for models of arithmetic.; In Set Theory, Arithmetic, and Foundations of Mathematics<. Eds. Juliette Kennedy & Roman Kossak. Lecture Notes in Logic (No. 36). pp.66-79. Cambridge University Press, 2011.
 Roman Kossak and James H. Schmerl. 2006. The structure of models of Peano arithmetic. Oxford Logic Guides, 50. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2006.