 Parameter-free definability

A definable set has parameter-free definability when its defining first-order formula uses only the original first-order language, with no additional constants naming elements. Constants already present in the language are allowed. Every such set is invariant under all automorphisms of a first-order structure. For example, zero in the constant-free additive integers is defined by $x+x=x$, while the singleton containing one is not parameter-free definable because negation is an additive automorphism moving it. Invariance is a necessary condition in every structure, and becomes sufficient for finite sets in a countable omega-saturated model.