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.

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.