Scott formula (${\varphi }_{\bar{a}}^{\alpha }(\bar{x})$)

A Scott formula describes a finite tuple at a transfinite stage of the back-and-forth method. Stage zero specifies the complete atomic description; successor stages specify matching one-element extensions in both directions; limit stages conjoin the earlier descriptions. For a countable structure the equivalences on its finite tuples stabilize at a countable ordinal.