In nonstandard analysis, the limitation to first-order analysis can be avoided by using a construction known as a superstructure. Superstructures are constructed in the following manner. Let X be an arbitrary set whose elements are not sets, and call the elements of X "individuals." Define inductively a sequence of sets with S_0(X)=X and, for each natural number k,

 S_(k+1)(X)=S_k(X) union P(S_k(X)),

and let

 S(X)= union _(k=0)^inftyS_k(X).

Then S(X) is called the superstructure over X. An element of S(X) is an entity of S(X).

Using the definition of ordered pair provided by Kuratowski, namely (a,b)={{a},{a,b}}, it follows that (a,b) in S_2(X) for any a,b in  X. Therefore, X×X subset= S_2(X), and for any function f from X into X, we have f in  S_3(X). Now assume that the set X is (in one-to-one correspondence with) the set of real numbers R, and then the relation R which describes continuity of a function at a point is a member of S_6(X). Careful consideration shows that, in fact, all the objects studied in classical analysis over R are entities of this superstructure. Thus, first-order formulas about S(X) are sufficient to study even what is normally done in classical analysis using second-order reasoning.

To do nonstandard analysis on the superstructure S(X), one forms an ultrapower of the relational structure (S(X), in ). Los' theorem yields the transfer principle of nonstandard analysis.

See also

Los' Theorem, Nonstandard Analysis, Ultrapower

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

Explore with Wolfram|Alpha


Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; and Lindstrøom, T. Nonstandard Methods in Stochastic Analysis and Mathematical Physics. New York: Academic Press, p. 16, 1986.Hurd, A. E. and Loeb, P. A. Ch. 3 in An Introduction to Nonstandard Real Analysis. New York: Academic Press, 1985.

Referenced on Wolfram|Alpha


Cite this as:

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

Subject classifications