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Unintended/Non-standard model/NSM/Field: Problem: we cannot simply say that the non-standard model is unintended.
Non-disquotational view: here it is only meaningful to speak of "unintended", if we can state by what facts about our practice these models are unintend - and precisely because these models make each of our sentences just as true, the specification of such facts appears to be impossible.
Applying/Explanation/Observing/Field: our observation practice explains how our physical vocabulary applies to all that and only that to which it applies to. - That explains why some non-standard models are unintended.
Unintended Model/Interpretation/Putnam/Field: there is nothing in our use of the set theoretical predicates that could make an interpretation "unintended". - (VsObjectivity of mathematics). - FieldVsPutnam: but this cannot be extended to the number theory.
Not every objective statement is formalizable. - E.g. Consequences with the quantifier "only finitely many".
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980