Relational databases embody data in tabular forms and show how certain
objects stand in certain relations to other objects. As an example
adapted from (Barwise 1990), the database in Figure 11
includes three binary relations: FatherOf, MotherOf, and BrotherOf.
(Binary relations can be represented as sets of ordered pairs such
that if an object a stands in relation R to another object b,
denoted by aRb, then 
.) A database
model is a function M with domain some set Rel of binary
relation symbols such that for each relation symbol 
,
is a finite binary relation that holds in model M.
Figure 11: A relational database consisting of three binary relations
If one wants to add a new relation symbol SizeOf to this database,
then 
SizeOf
. A database model M for 
is correct if the relation SizeOf
contains all pairs
where 
and n = |R|, the cardinality
of R. Such a relation can be seen in Figure 12. Now it
may be taken for granted that every database for Rel can be extended
in a unique way to a correct database for 
. Unfortunately, this
is not so.
Figure 12: The SizeOf relation defined for the database in Figure 11
Assuming the FA, it can be shown that there are no correct database
models. Because if M is correct, then the relation SizeOf stands in
relation SizeOf to n, denoted by SizeOfSizeOfn. But this is not
true in ZFC because otherwise 
SizeOf
SizeOf.
If Hyperset Theory is used as the meta-theory instead of ZFC in modeling such databases, then the solution of the equation
(which can be found by applying the Solution Lemma) is the desired SizeOf relation.