Definition:Relational Structure/Warning

Warning concerning Relational Structures
In the context of class theory, it is common to abuse notation by writing $\struct {C, \RR}$ when $C$ is a class and $\RR$ is a relation on $C$, and to call this a relational structure.

One must take care, as if $C$ is a proper class then it cannot be a member of any class.

By the set-theoretic definitions for ordered pairs, if $\struct {C, \RR}$ is an ordered pair then $C$ is a member of some class, which is a contradiction.

Thus, $\struct {C, \RR}$ is not a formal mathematical object of any kind, let alone an ordered pair, but only notational shorthand for a concept.