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Model theory is a general theory of interpretations of Axiomatic Set Theory. It is the branch of Logic studying mathematical structures by considering first-order sentences which are true of those structures and the sets which are definable in those structures by first-order Formulas (Marker 1996).
Mathematical structures obeying axioms in a system are called ``models'' of the system. The usual axioms of Analysis are second order and are known to have the Real Numbers as their unique model. Weakening the axioms to include only the first-order ones leads to a new type of model in what is called Nonstandard Analysis.
See also Khovanski's Theorem, Nonstandard Analysis, Wilkie's Theorem
References
Doets, K. Basic Model Theory. New York: Cambridge University Press, 1996.
Marker, D. ``Model Theory and Exponentiation.'' Not. Amer. Math. Soc. 43, 753-759, 1996.
Stewart, I. ``Non-Standard Analysis.'' In From Here to Infinity: A Guide to Today's Mathematics.
Oxford, England: Oxford University Press, pp. 80-81, 1996.