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.