In geometry, the term "enlargement" is a synonym for expansion.

In nonstandard analysis, let X be a set of urelements, and let V(X) be the superstructure with individuals in X:

1. V_0(X)=X,

2. V_(n+1)(X)=V_n(X) union P(V_n(X)),

3. V(X)= union _(n in N)V_n(X).

Let ^*:V(X)->V(^*X) be a superstructure monomorphism, with X subset= ^*X and ^*x=x for x in X. Then V(^*X) is an enlargement of V(X) provided that for each set A in V(X), there is a hyperfinite set B in V(^*X) that contains all the standard entities of ^*A.

It is the case that V(^*X) is an enlargement of V(X) if and only if every concurrent binary relation r in V(X) satisfies the following: There is an element y of the range of ^*r such that for every x in the domain of r, the pair (^*x,y) is in the relation ^*r.

See also


This entry contributed by Matt Insall (author's link)

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Gehrke, M.; Kaiser, K.; and Insall, M. "Some Nonstandard Methods Applied to Distributive Lattices." Zeitschrifte für Mathematische Logik und Grundlagen der Mathematik 36, 123-131, 1990.Gonshor, H. "Enlargements Contain Various Kinds of Completions". In Proc. 1972 Victoria Symposium on Nonstandard Analysis. New York: Springer-Verlag, pp. 60-70, 1974.Gonshor, H. "Enlargements of Boolean Algebras and Stone Spaces". Fund. Math. 100, 35-59, 1978.Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press, 1985.Insall, M. "Some Finiteness Conditions in Lattices Using Nonstandard Proof Methods." J. Austral. Math. Soc. 53, 266-280, 1992.Insall, M. "Geometric Conditions for Local Finiteness of a Lattice of Convex Sets." Math. Moravica 1, 35-40, 1997.Insall, M. "Nonstandard Methods and Finiteness Conditions in Algebra." Zeitschr. f. Math., Logik, und Grundlagen d. Math. 37, 525-532, 1991.Luxemburg, W. A. J. Applications of Model Theory to Algebra, Analysis, and Probability. New York: Holt, Rinehart, and Winston, 1969.Robinson, A. Nonstandard Analysis. Amsterdam, Netherlands: North-Holland, 1966.Robinson, A. "Germs." In Applications of Model Theory to Algebra, Analysis and Probability (International Sympos., Pasadena, Calif., 1967).Schmid, J. "Completing Boolean Algebras by Nonstandard Methods." Zeitschr. für Math. Logik u. Grundlagen der Mathematik 20, 47-48, 1974.Schmid, J. "Nonstandard Constructions for Join-Extensions of Lattices." Houston J. Math. 3, 423-439, 1977.

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Insall, Matt. "Enlargement." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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