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.
Doets, K. Basic Model Theory. New York: Cambridge University Press, 1996.Hodges,
W. A
Shorter Model Theory. New York: Cambridge University Press, 1997.Manzano,
M. Model
Theory. Oxford, England: Oxford University Press, 1999.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.
