Definition:Relational Structure/Warning
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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.
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.): $\S 10.1$