From FOM Digest, Vol 101, Issue 34,a post by Walt Read:
… the result of the work on non-standard (non-Euclidean) models of geometry was a recognition of the other models as equally valid. Eventually it was considered reasonable that even “the” universe might better be modeled by one of the non-standard models. Do we see N the same way? Is there a unique thing denoted by “the natural numbers”, accessible to us through intuition or however, with the non-standard models of PA being essentially artifacts of the formalization process? Or do the non-standard models have equal claim as models of “the natural numbers”? Might we at some time in the future see one of the non-standard models as better suited to our understanding as we learn more about natural numbers?
[Links and tags are mine]
Mathematics under the Microscope 
Dana Scott remarked in the FOM Digest:
Let us recall:
TENNENBAUM’S THEOREM: There is no recursive non-standard model of {PA}.
FEFERMAN’S THEOREM: There is no arithmetically definable non-standard model
of all true sentences of arithmetic.
In other words, there are no models just lying around in the closet
waiting to be used. The situation is so much different in Geometry.
By: Alexandre Borovik on May 28, 2011
at 4:47 pm