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.
