|Unintended Models, philosophy: a model results from a formula in logic, if its interpretation - the insertion of values instead of the free variables - gives a true statement. For axiom systems, one speaks of the set of models that the system allows to construct. The problem of the unintended models arises when a statement obtained in the system is indeterminate in one respect, so that in turn it allows different interpretations. See also indeterminacy._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
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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"._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
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