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Superstructure

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 be an arbitrary set whose elements are not sets, and call the elements of "individuals." Define inductively a sequence of sets with and, for each natural number ,

and let

Then is called the superstructure over . An element of is an entity of .

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

To do nonstandard analysis on the superstructure , one forms an ultrapower of the relational structure . 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)

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References

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.

Superstructure

Cite this as:

Insall, Matt. "Superstructure." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Superstructure.html